I have tried to google this but r- and k-selection aren't very search engine-friendly terms, and it doesn't help that there is an algorithm called 'cuckoo selection'.

A cuckoo abandons its egg after laying, and the offspring is raised by another species. Does this make it 'k-selection by proxy'? Or is it just a k-selector because it doesn't matter who is doing the caretaking for it to count as K?

It does matter who is doing the caretaking - cukoo birds can invest more into larger numbers of eggs, choice of egg laying site, and larger eggs since they do not invest in parental care.
On the surface, this seems like it would be straight r-selection.  But keep in mind that there are several species of nest parasites and they experience different selective pressures.
In at least some species, nest-parasite eggs hatch earlier and the chicks are larger than the non-parasitic eggs. That means more investment into egg nutrients for the cukoos, which is K-selection.
In some ecosystems, the cukoo might experience high host acceptance, in which case it might have selective pressure for larger high-quality eggs.  In other environments, there might be a high rate of host rejection of the parasitic egg.  In which case there could be strong selective pressure for a larger number of low-quality eggs.