Two demos with distributions, for the Beta and multivariate beta (Dirichlet).

Beta

The Beta has 2 scalar parameters, which we could put on both sides of the graph, and you drag them up and down separately, so there would be no interaction, which is too bad.
Or we could put one draggable circle in the middle, and have the distance to the bottom left and bottom right be the parameters! But then we'll have to subtract a certain amount (0.5) to make sure they're 0 when the circle is at the bottom of the graph.

Dirichlet

This is the Dirichlet for K=3, so it's on the 2-simplex. Clockwise from top left: α=(6, 2, 2), (3, 7, 5), (6, 2, 6), (2, 3, 4).
We want to drag 3 scalars, which are independent of each other. We could put three circles in the simplex corners that you can drag to resize?