The main theorem of section 2.9 of Bernstein, Zelevinsky "[Induced representations of reductive $\mathfrak p$-adic groups. I](http://www.numdam.org/item/ASENS_1977_4_10_4_441_0/)" gives a criterion for the existence of a non-zero intertwining operator between two parabolically induced representations. In your case the characters of the maximal split torus have to be conjugate under an element of the corresponding Weyl group. 

 You do not need to assume that $F$ has characteristic $0$.