Which parametric family could describe these asymmetric copulas? In my recent project I keep encountering asymmetric copulas like this:

However, most of the common parametric copulas I could find are symmetric. While there are some resources on asymmetric copulas I can't figure out if any of them would be a good match to describe the relationship of these random variables.

Those plots look like the sort of copulas you can get from Khoudraji's device.  Here are a couple of examples:
kc <- khoudrajiCopula(copula2 = gumbelCopula(4, dim=2), shape=c(0.5, 0.95))
scatterplot_mat(rCopula(500,kc))


kc2 <- khoudrajiCopula(claytonCopula(iTau(claytonCopula(), 0.95)), gumbelCopula(iTau(gumbelCopula(), 0.95)), shape=c(0.95, 0.6))
scatterplot_mat(rCopula(1000, kc2))


I don't know of any canned method for estimating this sort of copula from data, but you could try to maximize the pseudo-likelihood using a general function optimization algorithm.  The parameters are the the parameters of the two input copulas, along with the two shape parameters.  You'd have to try it with a several different input copulas and compare the fits you get.
However, that seems like a lot of work, with no guarantee that you'll get a usable result (e.g., the optimization could fail).  Is there some reason you can't use a nonparametric copula estimator, such as the TLL?  The vinecopulib package includes fitting routines for TLL copulas, so if it were me I'd at least try that before trying to cobble together a workable fit with something more exotic.
