We present a physical model that describes bright and dark snakelike similaritons propagating in an asymmetric dual-core waveguide amplifier by considering self-focusing and self-defocusing Kerr nonlinearities. The asymmetric dual-core waveguide is composed of two adjoining, closely spaced, upper and lower cores, in which the upper one is an active tapered graded-index waveguide with a long-period grating along the medium while the lower one is a passive step-index waveguide. For different choices of tapered index profile, novel exact analytical similariton solutions undergoing a self-compression are obtained. Our results show that these optical structures can be effectively controlled by adjusting the long-period grating and amplitude of the source. Besides, the stability and interaction of two neighboring bright similaritons are discussed by means of numerical simulations. These results may simulate the possibility of experiments related to tapered grating dual-core waveguide amplifiers.