We consider the ground-state properties of an impurity particle ("polaron") resonantly interacting with a Bose-Einstein condensate (BEC). Focusing on the equal-mass system, we use a variational wave function for the polaron that goes beyond previous work and includes up to three Bogoliubov excitations of the BEC, thus allowing us to capture both Efimov trimers and associated tetramers. We find that the length scale associated with Efimov trimers (i.e., the three-body parameter) can strongly affect the polaron's behavior, even at densities where there are no well-defined Efimov states. However, by comparing our results with recent quantum Monte Carlo calculations, we argue that the polaron energy is a universal function of the Efimov three-body parameter for sufficiently low boson densities. We further support this conclusion by showing that the energies of the deepest bound Efimov trimers and tetramers at unitarity are universally related to one another, regardless of the microscopic model. On the other hand, we find that the quasiparticle residue and effective mass sensitively depend on the coherence length ξ of the BEC, with the residue tending to zero as ξ diverges, in a manner akin to the orthogonality catastrophe.