Thank you for all the great responses!
Based on the reply from JewishGenner Sharon Frank, who observed - it
seems that the term "stroller" had at some point in time connoted a
"vagabond" or "loiterer" - I searched just on the term stroller and
see that the Merriam-Webster online dictionary shows one definition as
vagrant, tramp, so it would seem that these probably were men living
in the park.
From Sherri Bobish's post it sounds like there were lots of amenities
in the park, so I can see how it would have been an attractive place
for a homeless man.
I'll post if I find any new information.