To expand on the answers provided, this Möbius band cannot be made from a rectangular (or any shaped) strip of paper since paper is a (roughly) unstretchable material. Thus any Möbius band you could make from a strip of paper would have to be an isometry a plane strip (a developable surface) since it would preserve distances. A plane strip has Gaussian curvature zero since it is a plane, and by Gauss's Theorema Egregium, isometric mappings preserve Gaussian curvature.
It is in fact not true that a parametrization of a flat Möbius band is unknown. There is the parametrization due to Sadowsky which proved the existence of such a parametrization as early as 1930, and Wunderlich found a closed form parametrization in 1962 (the texts of both authors are in German, but there are translations in the Journal of Elasticity if you care to take a look). The shape that is unknown is the Möbius band of minimum energy i.e., the shape that is adopted if you make a paper strip and leave it to adopt its own shape. There have however been significant numerical investigations of such a shape (see Starostin and van der Heijden for example).
Here are the references for the translations:
Hinz, D. F. & Fried, E. Translation of Michael Sadowsky’s paper “An elementary proof for the existence of a developable Möbius band and the attribution of the geometric problem to a variational problem”. J. Elasticity 119, 3–6 (2015).
Hinz, D. F. & Fried, E. Translation and interpretation of Michael Sadowsky’s paper “Theory of elastically bendable inextensible bands with applications to the Möbius band”. J. Elasticity 119, 7–17 (2015).
Hinz, D. F. & Fried, E. Translation of Michael Sadowsky’s paper “The differential equations of the Möbius band”. J. Elasticity 119, 19–22 (2015).
Todres, R. Translation of W. Wunderlich’s “On a Developable Möbius Band”. J.
Elasticity 119, 23–34 (2015).
And the Starostin and van der Heijden articles:
Starostin, E. L. & van der Heijden, G. H. M. The equilibrium shape of an elastic
developable Möbius strip. PAMM Proc. Appl. Math. Mech. 7, 2020115–2020116
Starostin, E. L. & van der Heijden, G. H. M. The shape of a Möbius strip. Nat.
Mater. 6, 563–567 (2007).
There are several other more recent works on this topic, but these give a good background to the problem.