The edges are completely solved by swapping two edges at a time, using the algorithm RUR'URU2R'U. In this stage, we ignore what is happening to the corners.
The corners are then solved using a succession of 3-cycles which affect only the corners. This is possible since the corners are guaranteed to have an even parity.
This strategy can also be applied to the 4x4x4 cube. Here is just one example strategy:
All edges are attached to their corners by swapping two edges at a time. In this stage, we ignore what is happening to the centres.
The centres are then attached to the corners using a succession of 3-cycles which affect only the centres.
The resulting pseudo 2x2x2 cube is then solved using pseudo 2x2x2 moves.
Of course, we could improve the speed by mixing in some free-form/intuitive moves. But here, I am just demonstrating that methods can be designed that avoid running into parity problems.