We present a calculation of the connected-diagram contributions to the
first three non-trivial Mellin moments for the pion and kaon, extracted using local operators with up to 3 covariant
derivatives. We use one ensemble of gauge configurations with two
degenerate light, a strange and a charm quark ($N_f$=2+1+1) of
maximally twisted mass fermions with clover improvement. The ensemble
has a pion mass $\sim$260 MeV, and a kaon mass $\sim$530 MeV. We reconstruct the $x$-dependence of the
PDFs via fits to our results, and find that our lattice data favor a $(1-x)^2$-behavior in the large-$x$ region for both the pion and kaon PDFs. We integrate the
reconstructed PDFs to extract the higher moments, $\langle x^n \rangle$, with $4 \leq n \leq 6$.
Finally, we compare the pion and kaon PDFs, as well as the ratios of
their Mellin moments, to address the effect of SU(3) flavor symmetry
breaking.