I am trying to model a parallel plate capacitor (on die capacitor generated using metal plates.)
I know the capacitance per unit area, and metal sheet resistance. No knowledge of the dielectric.
I want to model this capacitor as a subcircuit element.
How do I model metal sheet resistance into the capacitor series resistance?
Also, I want to account for multiple frequencies, so I think the model needs to include multiple parallel branches.
This will become more involved than you probably think:
First of all, you need to define the geometry, because, clearly, the effective resistance depends on where how much current flows.
But modeling the plates as anything but a perfect conductor means that you can't assume the charge density to be the same all over plates – exactly because a charge has lost energy from the feed to the point it sits "on the inner surface". If you don't model that energy loss, then there's no metal resistance.
So, now you have an interesting differential equation, because the field that builds up defines the current flows, which give the paths along which you need to integrate the resistivity to get to an effective resistance. But: the field isn't homogenous, and you'll need to calculate the position of the charges!
That also means that your whole ESR, just like in any other real-world capacitor, depends on the frequency you observe it. Maxwell is here to safe (or ruin, depending on what you're hoping for) your day.
And things get ugly: you can't solve this thing without defining boundary conditions. That also means you can't solve this sensibly if you're assuming the E-field is perfectly normal to the plates, and doesn't exist outside of the gap: Your plate capacitor needs to also consider the edge effects.
You, honestly, can't solve that analytically in the general case. You might construct special capacitor geometries where you can (and it still won't be pretty) make sufficient simplifications ($\varepsilon_r$ between the plates much larger than outside, capacitor rotationally symmetric with feed all along the circular edge, things like that), but realistically, this is a case for finite element methods of EM simulation.
