Parallel algorithms to color interval graphs
I'm not really familiar with the topic, but with some searching there are some results.

For example, Efficient parallel algorithms on interval graphs by Das and Chen, and A simple optimal parallel algorithm for the minimum coloring problem on interval graphs by Yu and Yang both gave an algorithm with $O(\log n)$ time and $O(n)$ processors on the EREW PRAM model. Even for the chordal graphs (which is a superclass of the interval graphs) there is an algorithm with $O(\log^2 n)$ time and $O(n)$ processors on the CRCW PRAM model, in the paper Efficient parallel algorithms for chordal graphs by Klein. Maybe you can take a look at the papers and references, to make sure if there are any additional assumptions, like whether the interval representations are given or not.