  The Eichler-Shimura isomorphism describes a certain cohomology group with
coefficients in a space of polynomials by using holomorphic modular/cusp forms.
It determines a canonical decomposition of the corresponding de Rham cohomology
group associated to a specific oper on a Riemann surface. One purpose of the
present paper is to establish its analogue for opers in positive
characteristic. We first discuss some basic properties on the (parabolic) de
Rham cohomology groups and deformation spaces of $G$-opers (where $G$ is a
semisimple algebraic group of adjoint type) in a general formulation. In
particular, it is shown that the deformation space of a $G$-oper induced from
an $\mathrm{SL}_2$-oper decomposes into a direct sum of the (parabolic) de Rham
cohomology groups of its symmetric products. As a consequence, we obtain an
Eichler-Shimura-type decomposition for dormant opers on general pointed stable
curves by considering a transversal intersection of related spaces in the de
Rham moduli space.
