This is a bit complicated and involves the Medical Suite, RP Borough, the PA Commonwealth and the residents who have access to both sides of the alleyway behind the Medical Suite. I'll try an explain this correctly and I'd enjoy any Parkers who have more accurate details to pipe-in, please.
The mess the back-alley behind the Medical Building is sad and well know by everyone in RP as a dangerous place for traffic. Up to now there's little that can and has be done because there's rarely a car crash despite 10,000 near misses a week.
The baseline is that Rt 13 (Chester Pike) and Sellers Avenue are both PA State Highways. The State prescribes and controls the entrance onto and exit from State roadways and this prohibits anyone from popping in a convenient in/out driveway without the State's permission.
As I understand it, the Medical Suite has exit privileges to exit traffic from Rt 13, but they do not have "enter" privileges that would allow them to exit traffic onto Rt. 13. Once drivers leave Rt 13, their only exit is the dirt alleyway behind the Medical Suite, that runs from behind the Parker down to Swarthmore Avenue. It is the dirt alleyway that's been granted permission the enter/exit traffic onto Swarthmore and Sellers. Slight of an act of Congress, the current entry/exit privileges along Rt 13 (Chester Pike) and Sellers avenue will not change.
However you are right that the real issue here is the dirt alleyway behind the Medical Suite and who should maintain it and keep it free of snow. RP Borough has the obligation to maintain the dirty alleyway but doesn't have the obligation to pave it. Ridley Park is dotted with small unpaved, unnamed alleyways and they are considered shared, right of ways, for the residences. Unlike most alleyways,the municipal sewer runs directly under the alleyway behind the Parker and the Medical Suite. RP Borough has the responsibility to maintain the sewer and covering it presents other issues; namely cost, future maintenance and snow removal. The current thinking is the heaviest use of the alleyway comes from the Medical Suite's commercial traffic and they should pave it - but if they pave it, they'll have to maintain it and plow it.
At present there's no official Borough snow removal police on the alleyway behind the Medical Suite and no plans to pave it and no plans to change the existing exit policy - until there's 9,999 near misses one week; I hope not but statistics produce results.