There are about 60 new mutations per generation. Most of these are going to happen before the germline (i.e., during the divisions between the zygote and the formation of gametes).

Assuming these are randomly distributed in the genome, you have about a 60% shot of picking up one in an exome, which you can probably sequence for a few thousand dollars. You could probably do a low-pass whole-genome sequence for a bit more and get all 60; this would definitively tell you which parent produced which child.

You have differences in each set of twin-parents due to a small number of mutations in somatic and germ cells early on in cell division.

It would be necessary to sequence the parent's germ cells to determine the actual germline sequence, as any somatic mutation that occurred early on that only affected, say, the ectoderm, would not affect the mesoderm that gave rise to the gonadal ridge.

If there are differences found, and these same mutations are found in the somatic cells of the children (this time, any somatic cell), then statistically you can say that X child with Z mutation must belong to Y parent with Z mutation in the germline with very high confidence. If more than one mutation is found, then it is almost a statistical impossibility that X is not the child of Y.

This is due to the fact that the same somatic mutation occurring in the same position independently in two individuals in incredibly low, even in locations of high genetic variability, if you only have a sample size of two individuals.

So, there is great difficulty in making the determination (the standard test will not work), but you can make the determination with deep sequencing, assuming there are mutations to track.

A test of all children of the two couples would show up as if all children had the same two parents. Strictly genetically speaking, since it was two pairs of identical twins reproducing, all of those children would be (genetic) siblings, not (genetic) cousins.

Our modern paternity testing's sensitivity would show them as siblings, but technically, due to random mutation shortly after the zygote split and other environmental factors many sets of identical twins do have base-pair differences (although they would need much more in-depth analysis to catch than a simple paternity/maternity test would provide).

To clarify, the type of genetic test commonly used to determine paternity would not be able to tell them apart. A much more thorough (and expensive) test would probably be able to tell them apart, but it might require a huge number of genes to be tested.

Identical twins come from a single fertilization event so any inherited germline mutations will actually be present in both twins.

If there are mutations distinguishing identical twins, they would have had to have happened in the morula stage when most twinning events occur and at which point each twin is already multicellular. But these unique identifying mutations wouldn't be present in every cell of their body as they necessarily occurred at a multicelled developmental stage.

If you sampled multiple tissue types from all parents and were able to ID mutations that occurred in cell lineages that also lead to the relevant gonad tissues then you might be able to tell their children apart.

Further complicating matters, mutations that occur during meiosis in each twin-parent would not be traceable back to that parent because we would not find that mutation in any of their somatic tissues. It is possible with sequencing of sufficient depth (to accurately call SNPs) and broad (sequencing from tissues in the lineage that leads to germ cells, but not the germ cells themselves to find post-morula stage mutant populations) to determine the parental identity in this situation, but it isn't really because the parents "don't have identical genomes" but rather because there are multiple, slightly different genomes in an adult organism and the set of similar genomes in each twin-parent wont be quite the same.

Their consensus genomes though (avg sequence of all of their genomes) would probably be identical though. The way we do genome resequencing right now wouldn't do a good job attacking this kind of situation.