Started a bathroom remodel for my daughter and floor is partical board then tile backer board and then tile on top. This is all over 1/2” plywood. What is the best way to get the backer board up? It looks like hardi backer board and not the typical gray cement board. No screws are visible through the mud mixture that adhered the tile to the backer board.

If they did it right there should be screws, its likely you just can't see them. Im remodeling my bathroom and used a heavy duty 12" flat top screw driver and crowbar to pull up the old cement board and plywood. Backer board shouldn't to difficult to pry up but I'm not a professional contractor so you'll probably get better advice from someone else.

I'd pull the particle board up along with it.

Renting an electric demolition hammer for a day would be worth the money, the rental place should have the appropriate attachment. If the job is new enough that they used Hardi they probably used backerboard screws to put it down.

In a former life, I frequently ran into backer board put down with roofing nails.

I second removing the particle board.