I find it all very interesting (and sometimes in a way that can be related to humans).
It gets complex because often there is one of two options, but then there are a lot of those options.
For instance, one or both sexes take part in nest building.
One or both sexes take part in nest site selection.
One or both sexes may be polygamous (given the opportunity).
Some species start several nests and choose which is best (or multiple partners).
Where males share incubation there it's possible for females to lay in other nests.
One of the pair may be predated.
I'm sure there are other variables that i need time to think of, but i do think all options are possible, though maybe not recorded as yet, i would not exclude the possibility of two species feeding one nest of young (mixed species young or not).
One thing of note is the maternal/paternal instinct is strong in some individuals while being lacking in others (not a judgment on my part).
We may view the cuckoo as a bad parent because it plays not part in raising it's offspring, yet, it can lay a lot more eggs than it could possible raise, by spreading them around several other birds nests, thus giving it's young a better chance of success !