The InterFerence Channel with a Cognitive Relay (IFC-CR) consists of a classical two-user interference channel in which the two independent messages are also non-causally known at a cognitive relay node. In this work a special class of IFC-CRs in which the sources do not create interference at the non-intended destinations is analyzed. This special model results in a channel with two non-interfering point-to-point channels whose transmission is aided by an in-band cognitive relay, which is thus referred to as the Parallel Channel with a Cognitive Relay (PC-CR). We determine the capacity of the PC-CR channel to within 3 bits/s/Hz for all channel parameters. In particular, we present several new outer bounds which we achieve to within a constant gap by proper selection of Gaussian input distributions in a simple rate-splitting and superposition coding-based inner bound. The inner and outer bounds are numerically evaluated to show that the actual gap can be far less than 3 bits/s/Hz.