  In the class of the so called non-dynamic Fractional Obstacle Problems of
parabolic type, it is shown how to obtain higher regularity as well as optimal
regularity of the space derivatives of the solution. Furthermore, at free
boundary points of positive parabolic density, it is proven that the time
derivative of the solution is H\"{o}lder continuous. Finally, at regular free
boundary points, space-time regularity of the corresponding free boundary is
obtained for any fraction $s\in(0,1)$.
