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Parton's Alice weakens the two sides in a number of ways, one of which is
reducing the starting piece density from 50% to 25%. Adding a third board
and changing nothing else would reduce that starting density to under 17%.
Rather than add pieces on the other boards to bring up the starting
density, which blocks pieces traveling between boards, add pieces to the
starting board, where all the pieces are already. One way to do this:
expand the board[s] to 10x10, which gives us room for 30 pieces/side and 4
empty ranks between armies. Now, everyone will have their own opinion of
what those pieces should be, and you are all free to chime in, in fact,
you're invited.  

I'll start, unless someone has already beaten me to it, of course. :D
Add several short range pieces. An interesting class to use is the
inclusive compound pieces. This gives another choice. A piece that steps
one square and/or leaps 2 squares, in either order, can change boards
once, at the complete end of its move, or twice, once after each complete
step of the compound move. Leave this choice up to the player, and you've
made the game more interesting and those pieces a bit stronger.