Just to be sure that my wording was clear enough:
Let us agree on some names:
eye pieces (= 3x3x3 edges): the pieces seen through the holes - they are NOT centres
centres: triangular pieces with one sticker
corners: the pieces with two stickers (The name is influenced by the mental model of a 3x3x3 cube with 6 baby faces)
baby face: 4 centres + 4 corners around a vertex
Here is the corner parity:
We see an odd permutation of corners.
Baby face turned by 45 degrees:
Baby face turned by 90 degrees:
The corners are now in an even
permutation: we need a 3-cycle to solve them.
The centres are in an odd
permutation on that
Because the centres come in 12 pairs, we can involve another one of the same colour on a different baby face (e.g. the other pink centre) to solve the centres by pure 3-cycles.