Well, first of I already experienced that before. When I was still back in elementary I want to study fine arts, and when I reached high school, I want to be an Architect.
So when I reach college my grandmother want me to take Computer Science ( more on progamming ), I don't like it but I still enrolled it since my grandmother wanted it ( as a sign of respect, so I graduated with a degree in computer science )

On your end, you can choose any course you like, and you need to ask your dad politely too, that " you want to study graphic design" , graduated with a degree is really awesome. Plus applying for a job makes it easier.

Lastly I didn't study in any graphic school, I learned this through work experience and it push me to the extent, which made me curious and all the software that I use now are self-thought through out my career. And having friends in the same area / degree helps you mold better, and work experience will make you a better web, graphic designer, animator or illustrator.

I hope that helped you, if you have any more questions just hit me an email and I'll be happy to help. I don't know if I answered all your questions though ^_^