is size of maximum matching less than that of maximum k-parity k-set packing?
Consider the case $k=3$, and the following set system:
$$
(a_1,b_1,c),\ldots,(a_n,b_n,c) \\
(a'_1,b',c'_1),\ldots,(a'_n,b',c'_n) \\
(a'',b''_1,c''_1),\ldots,(a'',b''_n,c''_n)
$$
Every packing contains at most 3 sets, but $d=n+2$.