I know that it is still possible to place stones at Washington Park...provided one can actually locate the grave. It is relatively easy to find the recorded Section, Lot, and Grave...and the Sections are clearly marked although not all sections are accessible at present. Along Natural Bridge are sections 5 and part of 8, I think...Those are kept cut...but there isn't a clear way to tell the exact lots...

I talked with an official from Berkeley City Hall; he told me that I might be able to find out more information from local funeral homes that helped with the removals...

That might be a way for you to start as well. If you have the section, lot, and grave numbers you may be able to ask Randle and Sons or Austin Layne if they are able to help. If you don't have that information, I do plan to go to the STLGS this weekend to do research. I'll look up the information and post it here if you can send the name of your father.
vfc