I’m a 17 year old private pilot, and my friend and his girlfriend who are also 17 want to fly with me. It’s not clear whether they are asking their parents permission. If their parents didn’t want them to go and find out, would I, the pilot, get in trouble? Are there any regulations regarding this subject? Should I get their parents written consent before taking them flying? This question pertains specifically to the United States.
There’s no FAA regulation requiring that. That being said, if your passengers are minors, their parents have the ultimate say as to what they can and cannot do. It would be wise to make sure that your friends have permission from their parents for the activity prior to taking them as passengers.
If you are asking about FAA regulations, the answer is "no, there are no regulations on this subject". I've been involved in EAA Young Eagle Rides, CAP glider encampments, etc, and if there had been any FAA regulations bearing on a requirement for obtaining parental permission before giving young people rides in aircraft, someone would have mentioned it at some point. Not to suggest that parental permission wasn't obtained anyway, for other reasons-- but a specific FAA regulation on the subject would surely have been mentioned if it existed.
You don't need their parents explicit consent anymore then you would to give them a lift in your car, but you do need to ensure you are legally covered. As PiC you have legal responsibilities and liabilities so before you can take on the risks associated with carrying passengers you should check with whomever would be sued if they got hurt. If you are legally a minor, that would be your parents and/or whoever pays your insurance premiums. Also be a little careful about crossing state lines, as the laws relating to transporting minors over state lines vary from state to state and can be quite surprising.