The Parker Solar Probe will make its closest approach yet to the Sun

Of course it is completely evaporated before hitting anything that remotely resembles a surface.

Is there even a surface?

Yes, if you zoom out enough you see it. Like with ordinary objects.

Most ordinary objects have the property that the density of object mass has a very sharp gradient near some 2d surface that encloses a compact domain, and outside that is close to 0. However not identically 0, since eg the object is constantly releasing vapor of atoms of itself. If you zoom out enough out out of anything it looks like that, depending on how sharp you wish the gradient to be to call it a "surface".

For objects where the gradient at the boundary is not great relative to our size we would subjectively experience no surface when coming close eg to a cloud.

Does a galaxy have a "surface"? We can often also "clearly see" the edge of it...

How do you define density?

Really? I suspect this is a troll but, in the static case, roughly the ratio of volume of space to how much gravitational force said volume applies on other objects. Extrapolate correctly for qm/gr - thats an exercise left for the reader.

Depending on your scales etc you may wish to group different things into the category of “object”, eg a car would likely be be selected as a valid grouping of atoms as an object by most people in conversation wheres it is mostly empty space at the micro level, and has a bunch of very different densities (many oom) at diff volumes even at the macro scale (eg the air in the trunk vs the engine block).