  We present a calculus of pseudodifferential operators that contains both
usual parameter-dependent operators -- where a real parameter \tau\ enters as
an additional covariable -- as well as operators not depending on \tau.
Parameter-ellipticity is characterized by the invertibility of three associated
principal symbols. The homogeneous principal symbol is not smooth on the whole
co-sphere bundle but only admits directional limits at the north-poles, encoded
by a principal angular symbol. Furthermore there is a limit-family for
\tau\to+\infty. Ellipticity permits to construct parametrices that are inverses
for large values of the parameter. We then obtain sub-calculi of Toeplitz type
with a corresponding symbol structure. In particular, we discuss invertibility
of operators of the form P_1A(\tau)P_0 where both P_0 and P_1 are zero-order
projections and A(\tau) is a usual parameter-dependent operator of arbitrary
order or A(\tau)=\tau^\mu-A with a pseudodifferential operator A of positive
integer order \mu.
