Packing with Non-Congruent Regions

Continuing from the question of tiling with mutually non-congruent triangles (and in general, with convex regions) with same area and perimeter, can one think of packing with mutually non-congruent equal area 2D regions all of which form a one parameter family? For example, packing the plane with ellipses, all of same area but different eccentricity.

Can one say something general: (say) such a packing with mutually noncongruent equal area convex  regions belonging to a one parameter family will always be inferior to packing with copies of any one of those regions?

An analogous question can be asked about covering instead of packing.