It seems (to me) that parameters are also kind of variables in descriptions of physics. I am confused what makes a parameter different from a variable. Can anybody please explain this?
From a mathematical point of view, a parameter is a variable that is meant to stay fixed in a certain application. For example, in middle or high school you studied the three-parameter family of functions
f(x) = ax^2 + bx + c.
While the three numbers a, b, c are variables, you want to think of them as fixed and let only x vary. To distinguish the roles of a,b,c on the one side and x on the other, we refer to a,b,c as parameters.