I may be wrong, but I assume that once Twitter becomes a private company there will be no more board and Elon will be the CEO. Parag may stay at the company in a different role.

I don't think Elon particularly wants to be in the actual driver's seat of Twitter. That's a job and a half, especially when he already has SpaceX and Tesla to manage. I rather assume he will have someone competent at the helm to which he can give incredibly broad orders like 'figure out how to make free speech work'. Though that doesn't mean Parag won't be out - the competent person is likely to be someone who is not soft on free speech like Parag is, and there's a good chance it'll even be Jack.