We derive nonuniform four and six point subdivision rules for parametric curves based on local variational refinement. We stress the connection between the parametrization of the curves and the nonuniformity of the subdivision schemes. In particular, the nonuniform parameter sequences are subdivided in a way that balances out the spacing in the limit. To provide more shape control, we introduce ‘stretching’ (or tension) parameters. To estimate the (geometric) smoothness of the curves we extend the definition of Hölder regularity to piecewise linear parametric curves parametrized by arc length. In particular, we conclude that our four-point scheme is G 1 and our six point scheme without tension is G 2 .