Let $$F_q$$ be the field of size q and SL(n, q) be the special linear group of order n over the field $$F_q$$ . Assume that n is an even integer. Let $${\mathcal {A}}_i\subseteq SL(n,q)$$ for $$i=1,2,\dots , k$$ and $$\vert {\mathcal {A}}_1\vert =\vert \mathcal A_2\vert =\cdots =\vert {\mathcal {A}}_k\vert =l$$ . The set $$\{\mathcal A_1,{\mathcal {A}}_2,\dots ,{\mathcal {A}}_k\}$$ is called a k-cross (n, q)-Parsons set of size l, if for any pair of (i, j) with $$i\ne j$$ , $$A-B\in SL(n,q)$$ for all $$A\in {\mathcal {A}}_i$$ and $$B\in {\mathcal {A}}_j$$ . Let m(k, n, q) be the largest integer l for which there is a k-cross (n, q)-Parsons set of size l. The integer m(k, n, q) will be called the k-cross (n, q)-Parsons numbers. In this paper, we will show that $$m(3,2,q)\le q$$ . Furthermore, $$m(3,2,q)= q$$ if and only if $$q=4^r$$ for some positive integer r. We will also show that if n is a multiple of $$q-1$$ , then $$m(q-1,n,q)\ge q^{\frac{1}{2}n(n-1)}$$ .