It happened immediately after taming it due to a script or something. I would move and the pet would move like normal, but when I stopped the pet would sit down in statue form. Sadly, the pet stopped doing this after being resummoned.
I remember some of them did this on the beta. I wasn't aware that any still did, but I guess it's only temporary. It looks pretty cool when they do.
It's a shame they couldn't still do it on some sort of idle timer. Say, if you don't move for 20 seconds, it sits down. Same with other pets that have some sort of resting animation. Birds could land, etc.