OK. SSAO renders your scene into a texture using a replacement shader to figure out a depth map of some kind. And when doing so, it ignores everything that is transparent. And your particles are, in fact, considered transparent by Unity because their shaders are tagged to be transparent. So, this results in an occlusion texture that is looking like your particles wouldn't be there and that gets overlaid on your current view. Thus the strange look.

From the top of my head, there's no easy fix for this. Even if you would rewrite your particle shader to not be considered transparent any more will not help. This would only result in weird looking square occlusion borders around your particles because of the poly-billboards of the particles themselves.

I could, however, imagine some kind of additional post-process, using a second, grayscale rendering of your particles (with particles in white and background in pitch-black) and use that then to add on the SSAO texture to make areas overlaid by particles whit again. But that's not only tricky because of object depths, but might also be a bit hard on the graphics hardware, depending on what you plan to do. But it would solve the issue, I think.

One thing you could check is if the another of the replacement shader examples helps in this case, the soft particles :)