Examples of hardness phase transitions Suppose we have a problem parameterized by a real-valued parameter p which is "easy" to solve when $p=p_0$ and "hard" when $p=p_1$ for some values $p_0$, $p_1$.

One example is counting spin configurations on graphs. Counting weighted proper colorings, independent sets, Eulerian subgraphs correspond to partition functions of hardcore, Potts and Ising models respectively, which are easy to approximate for "high temperature" and hard for "low temperature". For simple MCMC, hardness phase transition corresponds to a point at which mixing time jumps from polynomial to exponential (Martineli,2006).

Another example is inference in probabilistic models. We "simplify" given model by taking $1-p$, $p$ combination of it with a "all variables are independent" model. For $p=1$ the problem is trivial, for $p=0$ it is intractable, and hardness threshold lies somewhere in between. For the most popular inference method, problem becomes hard when the method fails to converge, and the point when it happens corresponds to the phase transition (in a physical sense) of a certain Gibbs distribution (Tatikonda,2002).

What are other interesting examples of the hardness "jump" as some continuous parameter is varied?

Motivation: to see examples of another "dimension" of hardness besides graph type or logic type
I'm not exactly sure if this is the type of problem you were looking for, but the phase transition of NP-Complete problems is a (by now) well known phenomenon.  See Brian Hayes's articles "Can't Get No Satisfaction" about the 3-SAT phase transition and "The Easiest Hard Problem" about the Number Partition Phase transition, for some popular articles on the subject.

Selman and Kirkpatrick were first to show numerically that the phase transition for 3-SAT was when the ratio of clauses to variables was at around 4.3.

Gent and Walsh were first to show numerically that the phase transition for the Number Partition Problem happened when the ratio of bits to list length was about 1.  Later this was proved analytically by Borgs, Chayes and Pittel.

Travelling Salesman, Graph Coloring, Hamiltonian Cycle, amongst others, also appear to have phase transitions for a suitable parameterization of problem instance creation.  I think it's safe to say that it is a commonly held belief that all NP-Complete problems exhibit a phase transition for a suitable parameterization.