  In these proceedings, we summarize the main results from the first-ever
calculations of the chiral-even and chiral-odd twist-3 parton distributions,
$g_T(x)$ and $h_L(x)$, of the proton from lattice QCD. We use an $N_f=2+1+1$
ensemble of maximally twisted mass fermions with a clover improvement. The
lattice has a spatial extent of 3~fm, the lattice spacing is 0.093~fm, and the
pion mass is $260$~MeV. The matrix elements are obtained with a source-sink
time separation of 1.12~fm to control contamination from excited states. The
calculation is based on the quasi-PDF approach and employs three values for the
proton momentum: 0.83~GeV, 1.25~GeV, and 1.67~GeV. The lattice data are
renormalized non-perturbatively using the RI$'$ scheme, and the final results
are presented in the $\overline{\rm MS}$ scheme at the scale of 2~GeV.
Furthermore, we compute in the same setup the helicity, $g_1(x)$, and
transversity, $h_1(x)$, distributions, which are used to compare $g_T(x)$ and
$h_L(x)$ to their Wandzura-Wilczek approximations. For $h_L(x)$, we combine
results for the isovector and isoscalar flavor combinations to disentangle the
individual up- and down-quark contributions.
