For which primes p does x³ + y³ = p z³ admit non-trivial integer solutions?

The proposed characterization by $p \bmod 3$ is not correct as stated. A direct counterexample is $p=2$: $$1^3+1^3=2\cdot 1^3.$$ So primes congruent to $2 \pmod 3$ cannot be excluded. The examples previously given for $p=3$ and $p=13$ were arithmetic errors: $1^3+1^3=2$, not $3$, and $3^3+(-2)^3=19$, not $13$. Factoring in the Eisenstein integers $\mathbb Z[\omega]$ is relevant, but splitting or inertness of $p$ alone does not prove the claimed if-and-only-if statement. A complete characterization of the primes represented by this cubic equation requires additional local and global conditions. Until such a characterization is supplied with a reliable reference, the safe conclusion is that the original congruence claim is false, with $p=2$ as the smallest counterexample.