There was a thread a while back about this, but it didn't get many answers. The short version is that there's essentially nothing in the interior of the Arabian peninsula. No cities to conquer, very small and migratory population, no agriculture that could sustain a conquering army or long-term garrison, and no rivers that might help trade and transport.

It's worth pointing out that this applies only to the interior. The peninsula's coasts are much more hospitable, and indeed the Persians and Egyptians both ruled parts of the coasts.