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Starting out from a new description of a class of parameter-dependentpseudodifferential operators with finite regularity number due to G. Grubb, weintroduce a calculus of parameter-dependent, poly-homogeneous symbols whosehomogeneous components have a particular type of point-singularity in thecovariable-parameter space. Such symbols admit intrinsically a second kind ofexpansion which is closely related to the expansion in the Grubb-Seeleycalculus and permits to recover the resolvent-trace expansion for ellipticpseudodifferential oerators originally proved by Grubb-Seeley. Anotherapplication is the invertibility of parameter-dependent operators of Toeplitztype, i.e., operators acting in subspaces determined by zero-orderpseudodifferential idempotents.

Parametric pseudodifferential operators with point-singularity in the covariable