Who currently holds the rights to Crash Bandicoot?

Well Sony and Naughty Dog never owned the rights to that IP. It was owned by Universal Interactive, who had a 3 game development deal with Naughty Dog (still an independent studio at the time), as well deal with Sony for exclusive distribution on the Playstation. Both of these contracts expired after the first three games in the franchise and Universal took the franchise to other developers and across multiple platforms. I believe Universal Interactive eventually bought out by Activision, so the rights to the franchise are owned by Activision

Looks like there were rumours that Sony had reaquired the rights but Activision has refuted this so I’d go with Activision.

Thank you @Darth_Algar, very informative.