I ask this question mainly by curiosity.
See here for definitions and a plot of the clique numbers of the Paley graphs for the primes $p\equiv 1 \pmod 4$ up to $10000$.

• Is there an explanation why more clique numbers are odd than even?

The graphic below displays the proportion of even clique numbers among the first $n$ such primes, $n\le607$, which seems to stabilize around $20\%$.

• As $p$ grows, is anything known about the limit of this proportion?