We establish a multiparameter extension of the stochastic sewing lemma. This allows us to derive novel regularity estimates on the local time of locally non-deterministic Gaussian fields. These estimates are sufficiently strong to derive regularization by noise results for SDEs in the plain. In this context, we make the interesting and rather surprising observation that regularization effects profiting from each parameter of the underlying stochastic field in an additive fashion usually appear to be due to boundary terms of the driving stochastic field.