Are these sentences all acceptable in this context?
Do they have different meanings in this context?
Do '2' and '4' imply that all of them go to parks?
Yes, they are all acceptable in that context; yes, they have different meanings; and yes, (2) and (4) imply that all of them go to parks, though the hearer would probably be unlikely to hold the speaker responsible if there were discovered to be an exception or two, the idea being that basically all of them do.
I do find (1) and (2) to be more natural than (3) and (4) in the context. Reply (4) could, however, be made parallel to (2) if it were truncated to "The old people." Regarding (3), its meaning is much different from that of the others. It implies that the people the speaker is referring to should be old, whether or not they are.