  We present a parametric, grid-based lens model for the galaxy cluster MACS
J1149.5+2223, concentrating on the properties of the first multiply imaged
supernova Refsdal. This model complements our updated light-traces-mass (LTM)
strong-lensing model for this cluster, described in a companion paper, and is
generated using the same pipeline but with a different parametrization.
Together these two models probe different possible solutions in a relatively
self-consistent manner and can be used to examine systematic uncertainties and
relevant differences between the two parameterizations. We obtain reasonably
similar (agreeing to within $\simeq1-3\sigma$, in most cases) time delays and
magnification ratios, with respect to S1, from the two different methods,
although the LTM predictions seem to be systematically shorter/smaller for some
of the images. Most notably, the time delay [and 95\% CI] between the Einstein
cross (in particular, image S1), and SX, the image that appeared about a year
after the original discovery of the cross, differs substantially between the
parametric method (326 [300 -- 359] days) and the LTM method (224 [198 -- 306]
days), which seems to underestimates the true reappearance time. The cause for
this systematic difference is unclear at present. We speculate on its possible
origin and note that a refined measurement of SN Refsdal's properties should
help to more strongly discriminate between the two solutions, and thus between
the two descriptions for the intrinsic shape of the underlying matter
distribution. We also discuss the implications of our results for the Hubble
constant.
