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.

Depends.

What do you want the BMS to do?

When I design a BMS, the first thing it does is safety:

• Temperature monitor for charge/discharge
• Over current events (with a software fuse)
• Charge control
• stopping over discharge events

The only other tasks a BMS will do are things like:

• state of charge
• some basic functionality (if you're doing something very basic, complicated functionality should be taken out of the BMS to make sure the safety tasks are taken care of)
• cell balancing (if you have a series pack of cells)

So, as you have your charger outside the BMS, the question is how safe do you want you pack to be? If you're selling it, then you need a BMS as safety is (should be) paramount. If it's just for you, you know what you're doing, and you can control the environment, charge and discharge patterns and all that, then there is no reason to have any BMS at all.

Personally, I'd say yes you do. But if I'm quickly throwing something together, I have been known to go without.

You shouldn't plan on using the battery of 3 individual cells in parallel. You should use pre-assembled packs if you really need high-discharge current or better capacity. The pre-assembled packs are likely to contain cells with tightly matched characteristics. The manufacturer would select the cells from batches of thousands before making the assembly, which you have no luxury of having.

The major reason is that the weaker cell in parallel configuration will be subject to overcharging on each charge-up cycle, which would lead to accelerated irreversible damage to cell electrodes and/or separators. This is not due to "overcurrent" or "overvoltage", the charger has proper limits for these, but due to overexpose the weaker cell to charging voltage for longer than it should be, since the charger has no means to monitor how the current is split between cells and properly terminate the charge process. Unless you have current monitors and individual switches for each cell.