No, you didn't act inappropriately in seeking a recommendation, assuming all the facts are stated accurately. Why?
The fact that R, the recommender, is a friend of your parents is irrelevant. You are known to R who is in a position to make an honest evaluation of your potential for success. Had you, somehow, induced R to make an unfair evaluation then there would be an ethical issue, but that isn't what you suggest here.
The fact that R is a co-author with P, the professor, is meaningful, but that sort of thing is perfectly allowable. In fact, as long as R is honest (not under your control), P gets a recommendation from someone s/he is likely to trust.
In fact, the whole recommendation letter process depends on trust. You trust that R (whoever it is) knows about you and will report honest - hopefully good - things. P trusts R by reputation or otherwise and you chose R partly because you have a sense that P can trust R, whether they are colleagues or not.
Nothing in the current situation changes any of that except that, assuming R is honest, trust is actually enhanced.
What went wrong, however, is that P was embarrassed. That is unfortunate and it came back to haunt you. But, in the best of all worlds (not this one, I know), P should have been prepared for the possibility. But that is on R, not on you. R should have stated to P, and maybe did, that you were the child of a friend as well as your obviously great future potential.
There is no reason for R to lie, as it will lessen the relationship to P. There is no reason for P to accept you if you are actually not qualified unless the situation is that R is somehow forcing P to act against P's best interest. (Yes, this might be the case if R is especially powerful - but that is a different scenario).
I guess the lesson here isn't that you can't use friendship or other contacts in the admissions process, but that you shouldn't embarrass important people. Every candidate will use whatever relationships they have access to and it isn't a problem if people are honest.
In fact, if the child of a prominent academic wanted to follow in a parent's footsteps it would be impossible to gain entry if no friends of the parent could be recommenders, even if those friends were mentors of the child. S/he would have to rely on less informed advocates.
If you are, in fact, competent, then other students in your (previous) lab have no complaint since they, too, were accepted. They weren't disadvantaged in any way. They might look down on you, but that is their issue, and you might need to work to prove yourself.
Note that this answer depends on and assumes both honesty and on the lack of a severe power imbalance.