References:

Up to injectivity vs. trivial kernel issues, this amounts to the questions of why $H(R)\to (H/Q)(R)$ is surjective, but this follows from Bruhat-like decompositions (see the references in loc. cit.).

——

Of course, if one wants to avoid such general/abstract results, one can apply the ideas from the comments. Namely, if $L$ is the Levi factor of $P$, then for affine $X$ the natural map

$$H^1(X_\mathrm{fppf},L)\xrightarrow{\sim} H^1(X_\mathrm{fppf},P).$$

This is essentially because the difference between $P$ and $L$ is the (split) unipotent group $R_u(P)$ which is an iterative extension of $\mathbb{G}_a$’s, which (as being just the structure sheaf), have vanishing cohomology on affines (see loc. cit. for precise references).

But then, as Loren explains in the comments, when $G=\mathrm{Sp}(V,\psi)$ then the only possibility for $L$ is that it is of the form $\mathrm{GL}(W)\times \mathrm{Sp}(V’,\psi’)$ for some $W$ and some $(V’,\psi’)$. But, obviously for the general linear group, and by the example above for the symplectic group, both of these groups are special, and thus evidently so is their product.

References:

Here is a more general answer than that which I suggested in the comments.

Throughout the following let us fix a field $F$.

Definition (Serre): A linear algebraic group $G$ over $F$ is special if for every finite type reduced $F$-scheme $X$, any $G$-torsor $T\to X$ is trivializable Zariski (opposed to fppf) locally on $X$.

Example: For any symplectic space $(V,\psi)$ over $F$, the group $G=\mathrm{Sp}(V,\psi)$ is special. Let me sketch the proof. Symplectic vector bundles on $X$ (i.e., a vector bundle $\mathscr{E}$ together with a symplectic pairing $\omega\colon \mathscr{E}\otimes_{\mathscr{O}_X}\mathscr{E}\to\mathscr{L}$) form a stack for the fppf topology. As $G=\mathrm{Aut}(V\otimes_F\mathscr{O}_X,\psi\otimes 1)$ we see that $G$-bundles on $X$ correspond to isomorphism classes of symplectic bundles $(\mathscr{E},\omega)$ which are fppf locally isomorphic to $(V\otimes_F\mathscr{O}_X,\psi\otimes 1)$. Thus, it suffices to show such an isomorphism actually happens Zariski locally.

By a standard spreading out argument (or something more general like [Č1, Lemma 2.1]) it suffices to treat the case when $X=\mathrm{Spec}(R)$ for a local ring $R$. In this case both $\mathscr{E}$ and $\mathscr{L}$ are trivial, and so we’re essentially claiming that (up to isomorphism) the only symplectic pairing on $R\otimes_F V$ is $\psi\otimes 1$. This is a simple exercise which I leave to you. $\blacksquare$

Thus, the answer to your question is yes by combining this example with the following observation.

Proposition: If $G$ is a special reductive group, then $P$ is special for any parabolic $P$ of $G$.

Proof: Let $\mathcal{Q}$ be a $G$-torsor on some reduced finite type $F$-scheme $X$. As in the example we may assume that $X=\mathrm{Spec}(R)$ where $R$ is a local ring. By assumption $\mathcal{Q}\times^P G$ is trivializable. Thus, we are reduced to showing that the kernel of $$H^1(X_\mathrm{fppf},P)\to H^1(X_\mathrm{fppf},G)$$ is trivial. But, observe that $P_X$ is a parabolic group scheme of $G_X$, and so this always holds by the following lemma (which should be more well-known). $\blacksquare$

Lemma (see [Č2, §1.3.5]): Let $S=\mathrm{Spec}(R)$ where $R$ is a semi-local ring and $H$ a reductive group $S$-scheme. Then, for any parabolic subgroup $S$-scheme $Q\subseteq H$ the map $$H^1(S_\mathrm{fppf},Q)\to H^1(S_\mathrm{fppf},H)$$ is injective.

References:

[Č1] Česnavičius, K., 2015. Poitou–Tate without restrictions on the order. Mathematical Research Letters, 22(6), pp.1621-1666.

[Č2] Česnavičius, K., 2022. Problems about torsors over regular rings: With an appendix by Yifei Zhao. Acta Mathematica Vietnamica, 47(1), pp.39-107.

Here is a more general answer than that which I suggested in the comments.

Throughout the following let us fix a field $F$.

Definition (Serre): A linear algebraic group $G$ over $F$ is special if for every finite type reduced $F$-scheme $X$, any $G$-torsor $T\to X$ is trivializable Zariski locally on $X$.

Example: For any symplectic space $(V,\psi)$ over $F$, the group $G=\mathrm{Sp}(V,\psi)$ is special. Let me sketch the proof. Symplectic vector bundles on $X$ (i.e., a vector bundle $\mathscr{E}$ together with a symplectic pairing $\omega\colon \mathscr{E}\otimes_{\mathscr{O}_X}\mathscr{E}\to\mathscr{L}$) form a stack for the fppf topology. As $G=\mathrm{Aut}(V\otimes_F\mathscr{O}_X,\psi\otimes 1)$ we see that $G$-bundles on $X$ correspond to isomorphism classes of symplectic bundles $(\mathscr{E},\omega)$ which are fppf locally isomorphic to $(V\otimes_F\mathscr{O}_X,\psi\otimes 1)$. Thus, it suffices to show such an isomorphism actually happens Zariski locally.

By a standard spreading out argument (or something more general like [Č1, Lemma 2.1]) it suffices to treat the case when $X=\mathrm{Spec}(R)$ for a local ring $R$. In this case both $\mathscr{E}$ and $\mathscr{L}$ are trivial, and so we’re essentially claiming that (up to isomorphism) the only symplectic pairing on $R\otimes_F V$ is $\psi\otimes 1$. This is a simple exercise which I leave to you. $\blacksquare$

Thus, the answer to your question is yes by the following observation.

Proposition: If $G$ is a special reductive group, then so is any parabolic $P\subseteq G$.

Proof: Let $\mathcal{Q}$ be a $P$-torsor on some reduced finite type $F$-scheme $X$. As in the example we may assume that $X=\mathrm{Spec}(R)$ where $R$ is a local ring. By assumption $\mathcal{Q}\times^P G$ is trivializable. Thus, we are reduced to showing that the kernel of $$H^1(X_\mathrm{fppf},P)\to H^1(X_\mathrm{fppf},G)$$ is trivial. But, observe that $P_X$ is a parabolic group scheme of $G_X$, and so this always holds by the following lemma (which should be more well-known). $\blacksquare$

Lemma (see [Č2, §1.3.5]): Let $S=\mathrm{Spec}(R)$ with $R$ semi-local, and $H$ a reductive group $S$-scheme. Then, for any parabolic subgroup $S$-scheme $Q\subseteq H$ the map $$H^1(S_\mathrm{fppf},Q)\to H^1(S_\mathrm{fppf},H)$$ is injective.

References:

[Č1] Česnavičius, K., 2015. Poitou–Tate without restrictions on the order. Mathematical Research Letters, 22(6), pp.1621-1666.

[Č2] Česnavičius, K., 2022. Problems about torsors over regular rings: With an appendix by Yifei Zhao. Acta Mathematica Vietnamica, 47(1), pp.39-107.