Phase transition in parameter (and not temperature) for a classical system

What are other examples of classical models which exhibit such a phase transition in a parameter and not in temperature?

Formulated like this, it is difficult to answer, because there are far too many examples (the simplest example would be the first-order phase transition in the Ising model below the critical temperature as the magnetic field crosses $0$; see, for instance, Chapter 3 in this book).

There exist, in fact, constructions allowing you to build models in which the order parameter is given by an essentially arbitrary finite collection $(f_1,dots,f_n)$ of local functions! See Section 16.13 in Georgii's book for more on this.

Alternatively, if you write down a Hamiltonian depending on various parameters, whose set of ground states changes as these parameters are varied, then, under suitable assumptions, you can prove that the corresponding zero-temperature phase diagram is homeomorphic to the phase diagram at (small) positive temperatures. The standard way of doing this is via the Pirogov-Sinai theory (see Chapter 7 in this book).

The above results show how common phase transitions driven by a parameter different from the temperature actually are.

I feel, however, that you are interested in a more specific situation, but then you should be more explicit.

What is know for the Potts model (or FK/random cluster representations) in other dimensions than 2? Does one know that $m_{q,p_c(q)}to 0$ for $qto q_c$ from above?

Even for $q=1$ (Bernoulli percolation), it is not known in general that $m_{q,p_c(q)}=0$ (it is known when $d=2$ and when $dgeq 11$, I think). So, even the problem of determining the order of the phase transition for general values of $q$ is largely open above dimension 2. The only exceptions are $q=2$ (the Ising model, see this paper) and $qgg 1$. In the latter case, the transition is known to be first order (the first proof is due to Kotecký and Shlosman).

Note that, when $dgeq 3$, the phase transition is expected to be of first order for all $qgeq 3$. This has been proved for models with interactions of sufficiently long (but finite) range in this paper (see also this one); this even applies to the two-dimensional model, showing that the behavior of the planar (that is, nearest-neighbor) model is far from generic.