Special pleading

What it is

Applies a standard to everyone else but claims an unjustified exception for one's own case.

Example "Conflict-of-interest rules matter, except for my family's company."

Self-serving bias bias
We tend to excuse our own behaviour with exceptions that we would not grant others, which is self-serving bias in argument.
Tu quoque fallacy
Tu quoque points out the critic's inconsistency, while special pleading excuses one's own; both are about who is held to which standard.
Excessive self-regard tendency bias
Asking for an exception for ourselves comes naturally when we regard ourselves as special.
No true Scotsman fallacy
Both rescue a belief from counterexamples; no true Scotsman redefines the category, special pleading claims an exception.