  We consider the problem of finding, for two pairs $(s_1,t_1)$ and $(s_2,t_2)$
of vertices in an undirected graphs, an $(s_1,t_1)$-path $P_1$ and an
$(s_2,t_2)$-path $P_2$ such that $P_1$ and $P_2$ share no edges and the length
of each $P_i$ satisfies $L_i$, where $L_i \in \{ \le k_i, \; = k_i, \; \ge k_i,
\; \le \infty\}$.
  We regard $k_1$ and $k_2$ as parameters and investigate the parameterized
complexity of the above problem when at least one of $P_1$ and $P_2$ has a
length constraint (note that $L_i = "\le \infty"$ indicates that $P_i$ has no
length constraint). For the nine different cases of $(L_1, L_2)$, we obtain FPT
algorithms for seven of them. Our algorithms uses random partition backed by
some structural results. On the other hand, we prove that the problem admits no
polynomial kernel for all nine cases unless $NP \subseteq coNP/poly$.
