You have two problems. First, the form of your region. You have this:

ParametricRegion[{Cos[t]*(Sin[4 t] + 2), Sin[t]}, {t, 0, 2 \[Pi]}]

ParametricRegion isn't like Table where you have an iterator. That last term should be a list of variables. When you want to restrict the variables to a specific range, then you specify that range with its own list. So, you should have some thing like this:

ParametricRegion[{Cos[t]*(Sin[4 t] + 2), Sin[t]}, {{t, 0, 2 \[Pi]}}]

The second problem is that this correctly formatted region isn't a 2D region. It's a 1D region. It's just the boundary of the region you want the area of. So, if you take the area of the corrected ParametricRegion, you'll get Undefined