For parabolic PDEs, we present a new certainty equivalence-based adaptive boundary control scheme with a least-squares identifier of an event-triggering type, where the triggering is based on the size of the regulation error (as opposed to the identifier updates being triggered by the estimation error, or the control changes being triggered by the regulation error). The scheme guarantees exponential convergence of the state to zero in the L-2 norm and a finite-time convergence of the parameter estimates to the true values of the unknown parameters. The scheme is developed for a specific benchmark problem with Dirichlet actuation, where the only unknown parameters are the reaction coefficient and the high-frequency gain. For this specific problem, no existing adaptive scheme can handle the unknown high-frequency gain. An illustrative example allows the comparison with other adaptive control design methodologies. (C) 2019 Elsevier Ltd. All rights reserved.