The language gets a little confusing. The values in these equations which we're referring to were conceived of as "parameters", basically they are terms in the math that represent parts of the theory that need to be adjusted to match empirical observation, rather than terms whose values are dictated by the theory or by other terms. In that sense, they are variable, and hence parameters.
We call them "constants" because the value set by empirical observation is a single value. They are constant in the actual universe. But if you think of the mathematical model as a model that describes an entire class of metaphysically possible universes, then each hypothetical universe is described by a particular value (or set of values for different parameters in different models; i'm oversimplifying) of the parameter, which would be "constant" in that hypothetical universe, but which is variable from the standpoint of the model.
Edit: obviously everything in a mathematical model of physics is in some way set by observation, but it's like the difference in Newton's law of gravitation:
between G, a parameter in the sense we're talking about, and the main term which is the product of two masses divided by the square of the distance between them. m1m2/r^2 is the real "meat" of the model, G is this little fudge factor dictated by observation that sort of comes after the main insight into the force. All these fine tuning constants are sort of like G.