I recently bought an electric guitar for my son (aged 11).
I had originally insisted that he learnt on an acoustic guitar, but he rightly pointed out that an electric is easier to play for those with tender fingers because the action on an electric is usually lower and the neck is usually more narrow.
In the U.K. most music shops will do a package (guitar,amp,strap,lead) for under £200, but you must go and try the instrument to see if it feels right in your hands.
I bought a Squier Stratocaster (Fenders low cost alternative to the real made in the USA models) and it sounds just fine through a Marshall practice amp (Made in the UK)
Also, point out to your parents that an unplugged electric guitar makes very little noise, which can be a real bonus during those early days of practice.
Best of luck, David