I think my reasoning is correct, but I want to run through it here because having the right intuition will make similar problems easier in future.
A 2-simplex is homeomorphic to a closed disc, and a closed disc is homeomorphic to a hemisphere, so we can "build" $S^2$ out of two 2-simplices. However we need to have 3 specified vertices, say $v_0, v_1$ and $v_2$, on the circumference where the two hemispheres meet. This also means three 1-simplices joining these vertices.
Is this the simplest $\Delta$-structure possible?