How can I find the capacitance value changes that will result in linear tuning steps in a parallel LC circuit?
That depends on the frequency range you're targeting.  Frequency is a function of the inverse of the square root of capacitance, so you'll want to pick your endpoints first.  Your description above indicates zero capacitance is one endpoint, so find the maximum capacitance you'll support, and select the others to follow an X^2 curve.