I mentioned some of this in my reply to the schools thread as they are somewhat interlinked, but I think it's worth going into a bit more detail here.
In a word: NO. I think that they would have been very supportive had I learned a "proper" instrument and been interested in classical music, but I didn't and so they weren't.
My mum is very musical. She learned the violin at school, the guitar while she was doing her teacher training, has sung in some serious choirs and now in her 90s is part of and "oldies" ukulele group and probably did more "gigs" than me last year! My dad could play the piano, but I think that was just part of his standard upper-middle-class upbringing, he almost never played at home even though there was a piano in the house.
However both of them were very anti pop and rock music. Very little music was played at home, and for a long time the only record player we had in the house was a valve driven radiogram suitable mainly for playing 78s. They took me to a few classical concerts when I was young, which I found boring, although TBH I showed little interest in anything musical until...
In the summer of 1971 I spent two weeks at scout camp where Radio 1 was on all day every day and by the time I came home I was completely infatuated by the music of the nascent glam rock movement. My parents were horrified, and that is not over-exaggerating. While I was not actually forbidden to listen to pop music they did everything they could to make it as difficult as possible. The only record player/radio was in the living room and I could only use it when they weren't also in there which was almost all the time they were at home. I managed to buy a cheap radio off a friend at school which I could use in my room. They were also very discouraging about me buying records with my pocket money right up until I left home to go to university, although quite what they expected me to spend it instead I don't know, as by the time I was really into music I had pretty much abandoned all my other interests.
Considering my parents' attitudes to non-classical music and the fact that music was rarely on in the house, for some reason over the years we had acquired two guitars - a nasty cheap steel-strung instrument with massively high action probably bought from a catalogue, and a beginners classical guitar - as well as a reel-to-reel tape recorder. I took possession of the tape recorder as no-one else was using it and used it to tape my favourite songs off the radio. Then in 1973 my mum decided that she was going to attend "folk guitar" evening classes (she was doing teacher training and suspected it would improve her employment prospects) and when I asked if I could go as well, very surprisingly she said "yes".
For me the classes were a disaster. I had the steel strung guitar which was essentially unplayable and I hated the songs we are being taught to play. At the end of the year I could either hold down a chord or do some of the simpler finger-picking patterns, but not both together at the same time. However over the summer holidays I got hold of some sheet music and chord charts for music music I actually liked including The Beatles Complete song book, and suddenly I found I could actually play something recognisable. I badgered my parents to buy me a better guitar for my 14th birthday and was rewarded with a Kimbara branded steel-strung acoustic guitar, plus my own copy of the Beatles song book. This was the only time they positively helped me in my musical endeavours, although in buying me an acoustic guitar I think they hoped it would be a gateway into learning the Classical Guitar rather than "Rock n Roll". How wrong they were.
I quickly learned that I wasn't going to get any more help or encouragement from my parents, and if I was going to get an electric guitar I would have to do it myself. I sold all my model railway and some other toys and managed to get what I hoped was enough money for an electric guitar and small amp - two of my class-mates had managed to do this for under £40. However by the time I had the required funds, all those bargains had gone and I that all I could afford was just a solid electric guitar, or an amp and a cheap pickup for my acoustic. Deciding that an electric guitar without an amp wasn't much use I chose the latter. Unfortunately fitting a cheap acoustic guitar with a pickup still didn't make me sound like the guitarists on the records I was listening to, and although a home-made fuzzbox helped a little, it still wasn't right.
By then I had formed my first band with some friends form my school that shared similar off-beat tastes in music. We had a weird selection of musical instruments - a couple of acoustic guitars fitted with pickups along with my amp, no bass guitar although we were occasionally able to borrow one complete with a pathetically underpowered amp to add some dull, farty thuds behind the music; two "Bontempi" style organs that weren't in tune with each other, and a few home-brewed electronic circuits that produced weird noises when activated. Our "drum-kit" was composed of various tamborine and bongo style "drums" along with cans and anything else that made a suitable percussive noise when hit, all held together on a stands made from wooden dowels and clamps nicked from the school chemistry labs. As you can imagine, this was not a gigging band. Instead we composed and recorded hours of our own weird songs and instrumentals influenced by both prog-rock and the newly emerging post-punk DIY movement. However it was with this band and these ramshackle instruments that I recorded my first commercially released music and even got played on John Peel's radio show.
I had decided that the only way I was ever going to be able to afford an electric guitar would be to make one myself as I could buy the parts as I needed and could afford them. So for my last two years of school, when I wasn't in my 'A' Level subject classes I was in the woodwork shop slowly putting together my own solid electric guitar. I later worked out that I had spent more time doing this than any one of my actual exam subjects. At the time (late 70s) the only easily acquired information on making your own electric guitar was from International Musician magazine where Stephen Delft was producing instructions in monthly instalments. I quickly realised that I wouldn't be able to make my own truss-rod and therefore it would be best to buy a ready-made neck. and fit it to the body of my own design. I think the guitar ended up costing me just under £100 spread over about 18 months. The completed instrument was bought home at the end of my last term as a "fait accompli" and there was little my parents could do about it other than be disappointed in my choices, and by then I no longer cared what they thought. Besides I was likely to be off to university in October.
My parents continued to be disappointed with my choices regarding music when after 3 years of university I dropped out because it looked as though my band were about to be signed to a major record label (the label eventually decided that Wham! would be a better bet). My mum came to one of our gigs to see what all the fuss was about and absolutely hated it, didn't understand what we were doing, and told me as much.
However I think without all that parental disapproval I would never have been quite so determined to play and write music. Although I consider myself to be a reasonably good composer, even 45 years on I still have limited technical ability when it comes to playing guitar, bass or keyboards, and I still have work hard at many things that my band-mates find simple. So I do have to thank my parents for making it so hard, because otherwise I might have simply messed about on my electric guitar for a couple of years and then given it up for something newer and shinier like a lot of my other friends.
And ultimately my determination to play rock music has lead to my job. Back when I was a teenager, when I wasn't struggling to play the guitar and write songs, I was creating album covers for imaginary bands including Letrasetting complete lyric sheets. Later I learned to screen print so that I could produce impressive looking posters for the gigs I was playing. Eventually I got employment with a series of design and advertising agencies, until I went self-employed just over 11 years ago. These days I am mostly involved packaging design and artwork for chocolate products, although I still get to do the occasional record or CD cover.
Incidentally my sister who is completely and utterly tone-deaf had piano lessons and managed to pass all her grades until she got to the point where her inability to hold a tune or distinguish between two different notes could no longer be compensated by the fact that she could read music and play the right notes in the right order from it. The piano at home was old, didn't cope particularly well with 70s central heating and consequently was almost always noticeably out of tune. In particular the strings that produced G above middle C (which is seemingly in every beginners piano piece) were out of tune with each other, and that particular "note" came out as a horrible discord. She got plenty of encouragement from my parents despite the fact that her playing on our piano was musically jarring, because she was learning "proper" music.
TLDR: My parents gave be almost zero support and did much to try and discourage me, but as I approach my 60th birthday, it has made me the musician, composer and graphic designer that I am today.