The problem of failure of an adhesive contact between a parabolic (and also a cylindrical) indenter and an elastic half-space is considered. It is assumed that in the region where the adhesive contact has failed there is still frictional force determined by the Coulomb law. It is shown that two different scenarios of behavior are possible: either an abrupt tangential force decrease due to slipping or monotonic increase in the tangential force continued under further shift. A dimensionless parameter that distinguishes these two scenarios of behavior is defined