The parity of the number of quasigroups

Let I(n)I(n) be the number of isomorphism classes of quasigroups of order nn. Despite prior enumerations showing that I(n)I(n) is odd for 1≤n≤111≤n≤11, we find that I(12)I(12) is even. We also give a method for finding the parity of I(n)I(n), which we use to show that I(n)I(n) is odd for n∈{13,14,15,16,17,19,21}n∈{13,14,15,16,17,19,21}.