A parallel family of totally umbilical hypersurfaces in a Riemannian manifold is a smooth 1-parameter family of codimension -1 totally umbilical submanifolds every two of which are locally a constant distance apart. A perfect example is a family of concentric spheres in Euclidean space or some other space form. The most general examples are locally isomorphic to certain warped product spaces. Manifolds admitting such families are studied here by utilizing the canonical decomposition of the Riemann curvature tensor into three parts (the first being the Weyl conformal tensor, the second involving the Einstein tensor, and the third involving scalar curvature) and drawing out geometrical consequences from the assumption that one or another of these parts vanish when restricted to every member of the family. One result which is particularly interesting because this assumption is not visible in its statement is that the ambient manifold if conformally flat if and only if each hypersurface has constant sectional curvature. Other results characterize when the manifold is Einstein, Ricci-flat, etc. Differential equations for the metric and for the eigenvalues of the totally umbilical hypersurfaces are also developed showing in some circumstances that the ambient metric is determined by initial conditions on one of the hypersurfaces.