Disclaimer: I only have a small amount of experience in surface finish, so I am going from my engineering background and some theory.
Ra and Rz are measuring in the same axis, but are measuring different characteristics (Ra ~ average roughness, Rz ~ max roughness). If Ra increases, it makes sense that Rz must increase. However, Rz might increase without Ra increasing. Therefore, there should be a rough correlation between the two, but is will not be perfect. There is an article that discusses the correlation
here.
Now we get to Tp. Tp is measured on an axis that is 90 degrees to the axis in which Ra and Rz are measured. This means that Tp may vary significantly with little or no impact on Ra/Rz until you get to extreme values. You could get a weak correlation, but not one that is particularly useful, or that you would want to rely upon.