  The classical parabolic induction functor is a fundamental tool on the
representation theoretic side of the Langlands program. In this article, we
study its derived version. It was shown by the second author that the derived
category of smooth $G$-representations over $k$, $G$ a $p$-adic reductive group
and $k$ a field of characteristic $p$, is equivalent to the derived category of
a certain differential graded $k$-algebra $H_G^\bullet$, whose zeroth
cohomology is a classical Hecke algebra. This equivalence predicts the
existence of a derived parabolic induction functor on the dg Hecke algebra
side, which we construct in this paper. This relies on the theory of
six-functor formalisms for differential graded categories developed by O.\
Schn\"urer. We also discuss the adjoint functors of derived parabolic
induction.
