One place where (a generalization of) the desired result is stated explicitly is in a 1973 paper by J. H. Sampson, On a theorem of Chern. Sampson gives a simplified proof of Chern's result in his 1957 paper On a generalization of Kähler geometry (Algebraic geometry and topology. A symposium in honor of S. Lefschetz, pp. 103–121. Princeton University Press, Princeton, N. J., 1957).
The form in which Sampson states it is as follows:
Theorem: If a Riemannian manifold $(M^n,g)$ has holonomy $H\subseteq\mathrm{O}(n)$, then the $g$-Laplacian commutes with all of the linear operators on $\Omega^*(M)$ constructed from the ring of $H$-equivariant linear maps $L:\Lambda^*(\mathbb{R}^n)\to\Lambda^*(\mathbb{R}^n)$.
(Chern had proved this in his 1957 paper for the ring of operators $L$ as above that are degree preserving, but, in fact, this implies the more general result. To be more precise, both Chern and Sampson phrase the result in terms of $H$-structures without torsion, but, of course, this is the same as Riemannian manifolds with holonomy contained in $H$.)
The desired result is a special case of this, since, if $\pi$ is a $g$-parallel $p$-form, then the holonomy $H$ of $g$ is contained in the stabilizer of $\pi$, and hence, by the above result, the operator $L(\alpha) = \alpha\wedge\pi$ commutes with the Laplacian of $g$.
There are earlier results in special cases by André Lichnerowicz, e.g. Généralisations de la géométrie kählérienne globale. (French) Colloque de Géométrie Différentielle, Louvain, 1951, pp. 99–122. Georges Thone, Liège; Masson & Cie., Paris, 1951, but neither his articles nor those of Chern or Sampson mention Bott.