He is paying Tesla.

Most likely, the contract is between Tesla and Musk’s Twitter acquisition vehicle (“X Holdings”), so it isn’t moonlighting, it is their day job.

But, even if it is moonlighting - all employment contracts allow moonlighting with approval. Many firms are rather sparing in granting that approval - but it is a lot easier to say “yes” to an employee request if you know that’s what the CEO wants you to say.