NP-completeness of Induced disjoint paths
Your problem is NP-hard (and so NP-complete), by reduction from 3SAT.

Consider a 3SAT instances with variables $x_1,\ldots,x_n$ and clauses $C_1,\ldots,C_m$.

For each variable $x_i$, there are four vertices $a_i,T_i,F_i,b_i$, connected as follows: $a_i-T_i-b_i$ and $a_i-F_i-b_i$.

For each clause $C_j = \ell_{j,1} \lor \ell_{j,2} \lor \ell_{j,3}$ (where $\ell_{j,1},\ell_{j,2},\ell_{j,3}$ are literals), there are five vertices $c_j,L_{j,1},L_{j,2},L_{j,3},d_j$, connected as follows: $c_j-L_{j,k}-d_j$ (for $k=1,2,3$).
Additionally, if $\ell_{j,k} = x_i$ then we connect $L_{j,k}$ with $F_i$, and if $\ell_{j,k} = \bar{x}_i$ then we connect $L_{j,k}$ with $T_i$.

The source-sink pairs are $(a_i,b_i)$ (for $i=1,\ldots,n$) and $(c_j,d_j)$ (for $j=1,\ldots,m$).

Each $a_i-b_i$ path corresponds to a truth assignment of $x_i$, and each $c_j-d_j$ path corresponds to a choice of a satisfied literal in $C_j$.