  The quasi-PDF approach provides a path to computing parton distribution
functions (PDFs) using lattice QCD. This approach requires matrix elements of a
power-divergent operator in a nucleon at high momentum and one generically
expects discretization effects starting at first order in the lattice spacing
$a$. Therefore, it is important to demonstrate that the continuum limit can be
reliably taken and to understand the size and shape of lattice artifacts. In
this work, we report a calculation of isovector unpolarized and helicity PDFs
using lattice ensembles with $N_f=2+1+1$ Wilson twisted mass fermions, a pion
mass of approximately 370 MeV, and three different lattice spacings. Our
results show a significant dependence on $a$, and the continuum extrapolation
produces a better agreement with phenomenology. The latter is particularly true
for the antiquark distribution at small momentum fraction $x$, where the
extrapolation changes its sign.
