From a voltage standpoint, yes it could be done.
However, voltage is not everything. You need to look at the datasheet for the coin cells to see if they can supply enough current. In almost all cases (apart from some Li-Ion rechargeable ones), the rated continuous current sourcing capability is only ~2mA at most - typically only 1mA. Though it should be possible to run at a higher current at the expense of capacity - but probably not more than 10mA or so at a guess, depending on the battery.
With four in parallel, you can probably only supply 8mA to the board without significant voltage drop. The downside of running them in parallel is that if the batteries have slightly different terminal voltages, the lower voltage ones start acting as a load rather than a source).
By comparison, an AAA battery can typically source as much as 1A continuously, and also have much higher capacity.
Coin cells are designed for battery backup, or ultra-low power applications. They are not designed as a replacement for AAA batteries.
Having said all that, it is hard to tell what the requirements of the micro:bit are - in a quick look around I can't seem to find any information on power supply requirements/current draw. However looking at the processor on it, that requires at most around 11mA which may be within the capabilities of the coin cells. I'm not sure what else is on the board, however if you start trying to drive the LEDs, that will increase the current consumption far beyond the limits of the coin cells.
As a final note, you are getting your units mixed up - current and capacity are different things:
Current is a measure of the flow of electrons, with units of Amps [A] (mA = milliamp = 0.001 Amp).
Capacity is a measure of energy stored in the battery, and is measured in Joules, although frequently quoted in Amp-Hours [Ah] (mAh = milliamp-hours).