Do I need a BMS for Li-ion batteries connected only in parallel?
You do not need a full-blown BMS. Your stated charge/discharge currents are way below 18650 ratings (so you can get away without thermal sensor) and parallel cells do not need balancing.

But you do need a protection circuit. As a minimum a combination of discharge cut-off and discharge current limit. Current limit should be set slightly above your max 300 mA load, not the battery rating as typical protection circuits do.

The above assumes that your existing charging circuit has proper charge cut-off.

One missing thing here is automatic disconnect of a faulty cell. But not many BMS can do that anyway. It can be done with fuses.