Neither of these things seem particularly kosher moves to make. The question is how to figure out which one it is.
It would seem weird for Parag to be following Musk's orders given how Musk has behaved. It also seems weird for Musk to already have the insight into the company to know specifically who to fire. There's not much advantage to making these changes now.
On the other hand, going rogue and making big strategic decisions about the company really has the potential to burn Parag's reputation for wherever he would move next.
I guess there's a third option - that Musk has expressed a specific view, Parag has a different view, but that they both think that this move is necessary anyway so just got on and did it.