The intention and novelty in the presented study were to develop the regularity analysis for a parabolic equation describing a type of Eyring-Powell fluid flow in two dimensions. We proved that, under certain general conditions involving the space of bounded mean oscillation ( $BMO$) and the Lebesgue space $L^2$, there exist bounded and regular velocity solutions under the $L^2$ space scope. This conclusion was additionally supplemented by the condition of a finite square integrable initial data (also some of the obtained expressions involved the gradient and the laplacian of the initial velocity distribution). To make our results further general, the proposed analysis was extended to cover regularity results in $L^p\left(p>2\right)$ spaces. As a remarkable conclusion, we highlight that the solutions to the two dimensional Eyring-Powell fluid flow did not exhibit blow up behaviour.