Is getting an oral recommendation from a professor who has known me since childhood a conflict of interest during graduate admissions? My parents know a Professor A who knows another Professor B. Professor A has known me since I was a child. I applied for a Ph.D. position in Professor B's group and got admitted. Before admission, Professor A recommended me to Professor B orally but without writing a formal recommendation letter. I used the recommendation letters from other professors I really worked with.
Now, a Ph.D. student in our group learnt about this (my bad, I exposed this myself) and says I was cheating. And he just told everybody because of jealousy. I think I was fairly exploiting my personal network and also I don't know how much the oral recommendation was taken into account during admission. I also don't know what Professor A said to B. Whether he even really said something about me is a mystery to me. But I am totally qualified to get this position on my own.
Now many different versions of this rumor have come out. This is not the expected start of my Ph.D. life. I thought that everybody in my group might be friendly. But really, was I cheating during the admission? I might think about transferring to another group after all this.
I suspect @Buffy is right, that you did not do anything wrong. However, you did receive an advantage you did not earn because of who your parents are and who do they know. I suspect that that advantage is much more subtle and much more substantive than the oral recommendations (compare your experience with this answer Does politics and perception play a role in higher education? )
In short, you were lucky, and it is not moraly wrong to be lucky. However, it is understandable (not right, but understandable) that people who had to earn their position the hard way require extra evidence before they are willing to accept you as an equal. It is your challenge to deal with this situation gracefully. Demonstrate your competence, as Buffy suggested, but also learn how priviledged you are, and how challenging university can be for those who do not come from a middle class or higher background. A nice place to start is here: https://www.asanet.org/sites/default/files/attach/journals/feb15asrfeature_0.pdf