Solutions to easy Diophantine $8pq +1 = a^{2}$, p and q primes

Question

Show that $p = 3$ and $p = 5$ are the only primes with a maximal $3$ solutions each to $8pq + 1 = a^2$, where $p$ and $q$ are prime.

Answer 1

This isn't true. For example, with $p=11$, you can take $q = 5$ or $q=23$.