Totally nonparallel immersions

An immersion from a smooth n-dimensional manifold M into Rq is called totally nonparallel if, for every pair of distinct points x and y in M, the tangent spaces at f(x) and f(y) contain no parallel lines. The simplest example is the map R→R2 sending x to (x,x2). Given a manifold M, what is the minimum dimension q=q(M) such that M admits a totally nonparallel immersion into Rq? I will discuss how to apply Eliashberg and Gromov's "removal of singularities" h-principle technique to obtain existence results, and I will talk about some important considerations when studying differential conditions which manifest at pairs (or k-tuples) of points.