  We propose a new technique, which we call the lens parallax method, to
determine simultaneously the redshift distribution of the faint blue galaxies
and the mass distributions of foreground clusters of galaxies. The method is
based on gravitational lensing and makes use of the following: (1) the
amplitude of lensing-induced distortions of background galaxies increases with
redshift; (2) the surface brightnesses of galaxies decrease steeply with
redshift. The distortions of galaxy images due to lensing are thus expected to
be inversely correlated with surface brightness, allowing us to obtain relative
distances to galaxies as a function of surface brightness. If the redshifts of
the brightest galaxies are measured, then the relative distance scale can be
converted to mean galaxy redshifts as a function of surface brightness.
Further, by comparing the angular sizes of lensed galaxies with those of
similar galaxies in empty control fields, it is possible to break the so-called
mass sheet degeneracy inherent to cluster mass reconstruction techniques which
are based purely on image ellipticities. This allows an unambiguous
determination of the surface density of a lensing cluster. We describe an
iterative algorithm based on these ideas and present numerical simulations
which show that the proposed techniques are feasible with a sample of ~ 10 rich
clusters at moderate redshifts ~ 0.3-0.4 and an equal number of control fields.
The numerical tests show that the method can be used to determine the redshifts
of galaxies with an accuracy of dz ~ 0.1-0.2 at z ~ 1-1.7, and to measure the
masses of lensing clusters to about 5% accuracy.
