How to interpret parametric formulation of information bottleneck?
The notation stands for the maximum value of $-I_{\phi}(t;x) + \lambda I_{\phi,\theta}(t;y)$, where the maximum is taken over $\phi,\theta$. The result will be a function of $t,\lambda,x,y$.
Often in such a notation, the possible values of the variables being maximized over is restricted, and sometimes this forms part of the notation (say, $\max_{0 \leq \phi,\theta \leq 1}$). In other times, the possible values are restricted, but this is not reflected in the notation, and should be gleaned from context.