Is using the relatives’ friend’s recommendation inappropriate?
As a rule, where a significant personal connection exists between a referee and the subject, there is a potential conflict of interest. Definitions of "significant personal connection" are not universal. Policies on the acceptability and handling of potential conflicts of interest are also not universal. Where a potential or actual conflict of interest exists, it is imperative that it be declared.
A referee who has a significant personal connection to the subject may still be acceptable, if he/she has experience of working with the subject in a professional capacity, and the personal connection is declared. However, it should be understood that such a referee's comments would carry less weight than an equivalent referee (in terms of stature and experience of working with the subject in a professional capacity) without a significant personal connection. In some cases, it may be that such an "equivalent referee" may not exist, and so the referee with a significant personal connection is the only feasible choice (although, as I said in my answer to the related question, if more than one of the referees had a significant personal connection, that would be a red flag, since it would imply that the subject relied on friends to compensate for lack of merit or initiative).
Asking someone for a recommendation to whom you have a personal connection (like "parents' friend") is fine, provided the letter discloses the personal relationship and focuses on strictly factual professional matters. Such a disclosure will invoke suspicion in some readers, so this may not be the most useful letter, but submitting useless reference letters isn't unethical. (On the other hand, failing to disclose the personal relationship would be clearly unethical; even though it would be the writer's omission, it would still reflect badly on you.)
Where I think you slipped is in your public description of your references:
My friend and my parents’ friend both gave me a strong recommendation to my advisor.
Characterizing your letter-writers as "your friend" and "your parent's friend" strongly suggests that the recommendations could have been based on personal connections, rather than professional accomplishments, or at least that you thought the recommendations would be useful because of their personal connections to you. Compare your statement to the following, which describes exactly the same people in strictly professional terms:
My colleague and my undergraduate lab supervisor both gave me a strong recommendation to my advisor.
Even now in your StackExchange questions, you continue to describe your references primarily in personal rather than professional terms. For example, you "reached out to the friend" for a letter, not "asked your lab supervisor". That language raises suspicion.
So stop doing that.
And because your public description potentially raises suspicion about your recommendations, it also potentially raises suspicion about the people who judged your recommendation letters, which in this case means your advisor. If your letters are indeed suspect, then either your advisor is complicit in this unethical behavior, or your advisor was fooled by the suspect letters.
The situation is further complicated by the fact that your "parent's friend" is a coauthor of your advisor. That relationship creates yet another potential conflict of interest, albeit a mild one. (This is one of many reasons that PhD admissions decisions should not be made by a single person.)
As Buffy's answer says, this is all about trust and reputation, which require not only ethical behavior, but the appearance of ethical behavior. You are using language that suggests that you may have acted unethically; if so, then your advisor was either complicit or a victim. That suggestion—justified or not!—has the potential to undermine your advisor's reputation, or as Buffy put it, embarrassing him.
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.