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As has been noted, the answer is known for $\mathrm{GL}_2$. For $n>2$ there are certain cases where $\rho:=\mathrm{Ind}_{P(\mathbb{Z}/p^r)}^{\mathrm{GL}_n(\mathbb{Z}/p^r)}\pi$ is irreducible (see for example the result of Hill referred to in the question http://mathoverflow.net/questions/59305/parabolic-induction-for-gl2-z-pn/59364#59364).

For $\mathrm{GL}_3$ there are two papers by Campbell and Nevins (http://arxiv.org/pdf/0710.3261v1.pdf and http://arxiv.org/pdf/0710.3263.pdf) which study the decomposition of representation representations of the form $\rho$. In particular the authors almost achieve a decomposition of $\mathrm{Ind}_{B(\mathcal{O}_r)}^{\mathrm{GL}_3(\mathcal{O}_r)}\mathbf{1}$, but it turns out there are irreducible consituents which depend on the residue characteristic $p$, and these are hard to pin down explicitly in a uniform way.

Contrary to what's stated in the question, the simplest case is not when $P$ is a Borel but when it is a maximal parabolic subgroup of $\mathrm{GL}_n$. In this case one can find an explicit set of representatives for $P(\mathcal{O}_r)\backslash \mathrm{GL}_n(\mathcal{O}_r)/P(\mathcal{O}_r)$, independent of $p$, and consequently a complete decomposition of $\mathrm{Ind}_{P(\mathcal{O}_r)}^{\mathrm{GL}_n(\mathcal{O}_r)}\mathbf{1}$. This was done by Hill in "On the nilpotent representations of $\mathrm{GL}_n(\mathcal{O})$", and is also studied in a more general context in Onn & Bader (http://arxiv.org/abs/math/0404408).

In general, the problem of decomposing $\mathrm{Ind}_{B(\mathcal{O}_r)}^{\mathrm{GL}_n(\mathcal{O}_r)}\mathbf{1}$ will in some way or another involve a description of $B(\mathcal{O}_r)\backslash \mathrm{GL}_n(\mathcal{O}_r)/B(\mathcal{O}_r)$, and this is claimed to be a wild problem in Onn, Prasad & Vaserstein (http://arxiv.org/pdf/math/0506094.pdf), although Prasad seems to express a small degree of reservation in http://mathoverflow.net/questions/57922/bruhat-decomposition-for-gr-r-local-ring-or-r-mathbbz-mathfrakpr/57931#57931.

Looking at the results for $\mathrm{GL}_3$ it seems very likely that the problem to decompose of decomposing a general representation of the form $\rho$ is hopelessly complicated. As mentioned above there are partial results, and this is probably the most one can expect in general.

The "Iwahori decomposition plus the Bruhat decomposition over the residue field" approach alluded to in one of the comments seems unlikely to me to lead to a complete solution since it would reasonably be independent of $p$ while the decomposition problem in general probably depends on $p$ (because the spaces of double cosets do).

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As has been noted, the answer is known for $\mathrm{GL}_2$. For $n>2$ there are certain cases where $\rho:=\mathrm{Ind}_{P(\mathbb{Z}/p^r)}^{\mathrm{GL}_n(\mathbb{Z}/p^r)}\pi$ is irreducible (see for example the result of Hill referred to in the question http://mathoverflow.net/questions/59305/parabolic-induction-for-gl2-z-pn/59364#59364).

For $\mathrm{GL}_3$ there are two papers by Campbell and Nevins (http://arxiv.org/pdf/0710.3261v1.pdf and http://arxiv.org/pdf/0710.3263.pdf) which study the decomposition of representation of the form $\rho$. In particular the authors almost achieve a decomposition of $\mathrm{Ind}_{B(\mathcal{O}_r)}^{\mathrm{GL}_3(\mathcal{O}_r)}\mathbf{1}$, but it turns out there are irreducible consituents which depend on the residue characteristic $p$, and these are hard to pin down explicitly in a uniform way.

Contrary to what's stated in the question, the simplest case is not when $P$ is a Borel but when it is a maximal parabolic subgroup of $\mathrm{GL}_n$. In this case one can find an explicit set of representatives for $P(\mathcal{O}_r)\backslash \mathrm{GL}_n(\mathcal{O}_r)/P(\mathcal{O}_r)$, independent of $p$, and consequently a complete decomposition of $\mathrm{Ind}_{P(\mathcal{O}_r)}^{\mathrm{GL}_n(\mathcal{O}_r)}\mathbf{1}$. This was done by Hill in "On the nilpotent representations of $\mathrm{GL}_n(\mathcal{O})$", and is also studied in a more general context in Onn & Bader (http://arxiv.org/abs/math/0404408).

In general, the problem of decomposing $\mathrm{Ind}_{B(\mathcal{O}_r)}^{\mathrm{GL}_n(\mathcal{O}_r)}\mathbf{1}$ will in some way or another involve a description of $B(\mathcal{O}_r)\backslash \mathrm{GL}_n(\mathcal{O}_r)/B(\mathcal{O}_r)$, and this is claimed to be a wild problem in Onn, Prasad & Vaserstein (http://arxiv.org/pdf/math/0506094.pdf), although Prasad seems to express a small degree of reservation in http://mathoverflow.net/questions/57922/bruhat-decomposition-for-gr-r-local-ring-or-r-mathbbz-mathfrakpr/57931#57931.

Looking at the results for $\mathrm{GL}_3$ it seems very likely that the problem to decompose a general representation of the form $\rho$ is hopelessly complicated. As mentioned above there are partial results, and this is probably the most one can expect in general.

The "Iwahori decomposition plus the Bruhat decomposition over the residue field" approach alluded to in one of the comments seems unlikely to me to lead to a complete solution since it would reasonably be independent of $p$ while the decomposition problem in general probably depends on $p$ (because the spaces of double cosets do).