To my knowledge, it is not known whether $Ind_{P_{n,k}}^{GL_n(o)} 1$ decomposes with single multiplicity.

This is certainly necessary by Frobenius reciprocity $$ dim Hom_{P_{n,k}}( 1 , Res_{P_{n,k}} \pi) = dim Hom_{GL_n(F)}( Ind^{GL_n(F)} Ind_{P_{n,k}}^{GL_n(o)} 1, \pi).$$

For more information, see Parabolic induction GL(n,Zp)

To my knowledge, it is not known whether $Ind_{P_{n,k}}^{GL_n(o)} 1$ decomposes with single multiplicity.

This is certainly necessary by Frobenius reciprocity $$ dim Hom_{P_{n,k}}( 1 , Res_{P_{n,k}} \pi) = dim Hom_{GL_n(F)}( Ind^{GL_n(F)} Ind_{P_{n,k}}^{GL_n(o)} 1, \pi).$$

For more information, see Parabolic induction GL(n,Zp)