Sometimes parents seem to want to live vicariously through you. My Mum wanted (still wants) me to live my life how she would have is she didn't get married at 18 and had 4 kids! She just didn't get it, I didn't want to be a hairdresser and to sell crap on party plan to my 'girlfriends'!
And when I was 16 and I had a nose ring, a tattoo (no they don't wash off!) and listened to the beastie boys, hole and sonic youth I was perpetually grounded and clothes were chosen for me to wear whenever there was a rellie bash.
:/
You just got to be comfortable and sure of who you are. I know, easier said than done. If you parents don't like the mountain bikes and don't respect that you can't change their minds because mind control isn't right. Just let it be known mountain bikes make make you happy, are good for you (after 3 months of riding at least 3 times a week this will be physically evident) and if they get grouchy at you and the bikes start talking about sustainable transport policy, the overuse of cars to commute and chuck a few statistics their way.
That ought to shut them up, statistics silence grumpy folk.
I hope this makes you feel a little less frustrated.