Sonic Adventure Pose - Joke N64 Magazine ad

Perhaps, but Playstation was so new, I don't think anyone could have predicted them more or less taking over Sega's spot in the console lineup.

By contrast, Sega had endured as Nintendo's rival for quite a few years, so there was a history to exploit. Sega had a definite mascot Nintendo could mock, Playstation does not.

I'm referring to strictly * company-owned* mascots, own by them solely. Nintendo owns Mario. Sega owns Sonic.

PlayStation does not solely own any of those characters you just listed, except Toro Inoue. They merely published them on their platform.

Crash and Lara has recently been on many platforms, including PC, so they aren't even console exclusives any more. Twisted Metal hasn't had a game since 2002. Parappa hasn't had a game since 2001.

Only Toro is own soley by Sony and he's not very popular outside of Japan. So no, there is no globally well-known mascot control and owned solely by PlayStation.