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.