We consider a fermion-antifermion pair interacting via Dirac oscillator coupling in one-spatial dimension and obtain an exact energy spectrum for such a composite system by solving the corresponding form of a fully-covariant many-body Dirac equation. This energy spectrum shows that the formed composite system behaves as a single quantum oscillator carrying total rest mass of the particles. This result allows to analyse thermal properties of the considered fermion-antifermion pair. To acquire this, we study the thermal properties of such a pair in the framework of the theory of superstatistics where the probability density $f(\beta)$ follows three distributions: Gamma, Log-normal and F-distributions. First of all, we determine the partition function under the approximation of the low-energy asymptotics of superstatistics through the Euler-Maclaurin Formula. By using the desired partition function, we determine all of the thermal properties for such a pair in terms of the parameter $q$ and discuss the results.