Your parents are going to continue to interfere in your doings as long as you live under their roof. In fact, I have seen parents continue to assert that they had a controlling interest in their children's lives even after the child had moved out... in most cases it was because these parents were offering "help" with school, rent, groceries, or car repairs and felt that the expense on their child's behalf earned them a say so in how their child lived their life, choices they made, etc...
IMHO you can be dependent, or independent. There is no middle ground. (How very Yoda... ). I have known people who's parents were interfering to the point that when the phone rang (this is after the child had moved out and gotten their own place) and the parent's voice was heard, the child had to say "Unless this is about weather or sports, then shut up and leave me alone". Because everything that parent did was to try to snoop into what was going on in the child's life so they could find a way to interfere, offering "help" that became bribes that became an "obligation" to do what the parent dictated. It's sad and unfortunate when things come to that point, but a strong person will forge their own way and develop the independence necessary to assert their own freedom. In some cases this requires a level of discipline just short of obtaining a restraining order. It's really hard sometimes to get them to relinquish control, and impossible if one continues to accept "offers of help".
Good luck in dealing with your parents. It sounds like they consider themselves a co-owner of the dog with you... of course, even if you are giving the dog it's room and board, who's giving YOU room and board? Your parents? If that's the case, one can see why they might feel entitled to an equal or even a superior authority over the dog than yourself. Bears pondering.