In several places, such as [1] and [2], it seems to be implicitly known that (normalized) parabolic induction corresponds to tensoring affine Hecke algebras. I say implicitly because both [1] and [2] state Zelevinksy's classification theorem for representations of $GL_n$ in terms of affine Hecke algebras, despite the fact the paper of Zelevinksy they cite works purely with representations. I can't seem to find a source for this, and to me it is not obvious.

To be more precise, I'll first fix some notation. Let $F$ be a non-Archimedean local field, $G=GL_n(F)$, with Levi subgroup $M$, and respective Iwahori subgroups $\mathcal{I}_G$ and $\mathcal{I}_M$. We write the "full" Hecke algebras as $\mathcal{H}(G)$ and $\mathcal{H}(M)$, and the subalgebras of $\mathcal{I}_G$ (resp. $\mathcal{I}_M$) bi-invariant functions as $\mathcal{H}(G, \mathcal{I}_G)$ (resp. $\mathcal{H}(M, \mathcal{I}_M)$). Both [1] and [2] offer presentations of affine Hecke algebras $\mathbb{H}_G$ and $\mathbb{H}_M$, and we fix *the specific isomorphisms* $\mathcal{H}(G, \mathcal{I}_M)\to \mathbb{H}_G$ and $\mathcal{H}(M, \mathcal{I}_M)\to\mathbb{H}_M$, which I believe are originally due to Iwahori and Matsumoto.
Recall that there's a standard equivalence between $G$-reps with non-zero $\mathcal{I}_G$-fixed vector, and $\mathcal{H}(G, \mathcal{I}_G)$-modules. Via the isomorphism(s) of Iwahori-Matsumoto, we obtain an equivalence of $\mathbb{H}_G$-modules.

The theorem that seems to be made implicit use of in [1] and [2] is: Let $\sigma$ be a smooth representation of $M$ with non-zero $\mathcal{I}_M$-fixed vector, and let $\bar{\sigma}$ be the $\mathbb{H}_M$-module obtained through the above equivalences. Then $\mathbb{H}_G\otimes_{\mathbb{H}_M}\bar{\sigma}$ is isomorphic as an $\mathbb{H}_G$-module to the $\mathbb{H}_G$-module obtained from the normalized parabolic induction $\iota_{M\subset P}^G(\sigma)$.

While it may not be terribly deep to try and prove this, I fear aspects of it are tedious, and I'm hoping it has been carried out somewhere in detail.

The closest thing I could find is Theorem 1.4 of [3], which compares compact induction from closed subgroups to tensor products over the full Hecke algebras $\mathcal{H}(G)$ and $\mathcal{H}(M)$. Presumably, replacing these algebras with the $\mathcal{I}_G$ and $\mathcal{I}_M$ bi-invariant functions still leaves you with an $\mathcal{H}(M, \mathcal{I}_M)$-action on $\mathcal{H}(G, \mathcal{I}_G)$, and that somehow forces compact induction to become normalized parabolic induction. Then, one would need to transport everything via the isomorphisms with $\mathbb{H}_G$ and $\mathbb{H}_M$, which I think is rather tedious. For example, it is far from clear that the action of $\mathcal{H}(M)$ on $\mathcal{H}(G)$ described in [3], corresponds to the action of $\mathbb{H}_M$ on $\mathbb{H}_G$ (implied) in [1].

[1] D. Barbasch, C. Ciubotaru, Ladder representation of $GL(n, \mathbb{Q}_p)$, Representations of reductive groups, pages 117 -- 138

[2] B. Leclerc, M. Nazarov and J.-Y. Thibon, Induced representations of affine Hecke algebras and canonical bases of quantum groups

[3] P. Cartier, Representations of $\mathfrak{p}$-adic groups: A survey