Don't worry about it.
PETA uses some very questionable methods, and they're not a group you want to get mixed up in. But sending them a couple of emails won't get you in any trouble at all. They have no motivation to harm you because they won't get anything out of it.
Lots of people ask them about their stance regarding medication and medical research. If you even get a response, it will likely be a cut-and-paste stock response that they send to everyone who asks the same kinds of questions.
I'm not sure why people think PETA represents vegans in any way. They are NOT a group that represents the interests or needs of everyday vegans. Some beginning vegans and vegetarians look to them as a resource for information while forming their own views, but mostly PETA is a publicity machine. I'm a vegan and I make my own decisions. I don't do whatever PETA tells me to do!
You're not a researcher, and you're not a pharmacist - you're a student who is considering becoming a pharmacist or pharmacy technician in the future. You're not someone they're concerned about. They have no reason to bother you, keep information about you, or contact you at all.
They don't have real personal information about you. There are probably dozens of people in the country who share your name - I know there are dozens who share mine. Your email address is not that personal - you could and should have more than one. Just be careful in the future. For your general safety, never share personal info like your address, date of birth, and especially things like passwords, social security number, and mother's maiden name with anyone unless it is appropriate and necessary (like on tax and banking forms.)
Your dad wants to protect you, and that is exactly his job as a dad. He loves you and he's just being a good dad. But PETA can't hurt you. Animal rights activists don't actually ruin scientists' or doctors' lives. That's a misconception. They make a lot of noise in an effort to upset and offend people, but they can't ruin your future.
So dust yourself off, treat this as a learning experience, and keep studying what you want to study!