my advice would be to go with your parents' advice and get the acoustic first. It's the best schooling you can get and you won't annoy anybody too much in the first months when your technique will... ummmm... suck royally.
An electric won't hurt your fingers but it isn't the best place to start. it's more expensive to equip yourself (axe, amp etc) and it will sound awful at first.
Learning to play acoustic, with a wider neck, heavier strings and more difficult fretting technique than an electric will stretch your fingers, give you good strength and flexibility.
I first learned on a £50 acoustic that had a neck that seemed to be as wide as a telegraph pole but I mastered that and when I moved to electric with a flat neck and 9 gauge strings everything just seemed so simple and easy.
Also the best advice I can give is walk before you run. Some of the music you like is extremely complicated and imagining that you're going to crank out Magnum Opus in a couple of months is over-ambitious.
Your best bet is to buy your acoustic and at the same time nab yourself a songbook (with chord windows) of some simple enough songs but ones you like. Try some Beatles, Bowie, whatever. If nothing else these will teach you the fundamental of song construction, after all these guys were masters of the art of songwriting.
Also ther's millions of great solely acosutic songs out there, from Zep to Yes to Rush to Genesis to whoever... ti's not a limiting instrument. And open tunings will expand your horizons even further.
At the end of the day it's practice that will get you there. Don't get donwhearted when you find yourself with blistered fingers and no discernible improvement after a month. It takes time and effort.
But it is the most rewarding and enjoyable thing you will ever teach yourself. And you can only get better at it.
I've been playing for nearly 20 years and I still learn new things every day and it still thrills me and soothes me in equal measure as much as it did the very first day I ever picked the thing up.
It's an astonishing, always fascinating instrument and it will enrich your life.
Plus, if you're good, it'll get you girls.
isn't that the reason we all started to play?