We initiate a systematic study of algorithms that are both differentially private and run in sublinear time for several problems in which the goal is to estimate natural graph parameters. Our main result is a differentially-private $(1+ρ)$-approximation algorithm for the problem of computing the average degree of a graph, for every $ρ>0$. The running time of the algorithm is roughly the same as its non-private version proposed by Goldreich and Ron (Sublinear Algorithms, 2005). We also obtain the first differentially-private sublinear-time approximation algorithms for the maximum matching size and the minimum vertex cover size of a graph. An overarching technique we employ is the notion of coupled global sensitivity of randomized algorithms. Related variants of this notion of sensitivity have been used in the literature in ad-hoc ways. Here we formalize the notion and develop it as a unifying framework for privacy analysis of randomized approximation algorithms.