We give new sublinear and parallel algorithms for the extensively studied problem of approximating n-variable r-CSPs (constraint satisfaction problems with constraints of arity r up to an additive error. The running time of our algorithms is O(n/ε^2) + 2^O(1/ε^2) for Boolean r-CSPs and O(k^4 n / ε^2) + 2^O(log k / ε^2) for r-CSPs with constraints on variables over an alphabet of size k. For any constant k this gives optimal dependence on n in the running time unconditionally, while the exponent in the dependence on 1/εis polynomially close to the lower bound under the exponential-time hypothesis, which is 2^Ω(ε^(-1/2)). For Max-Cut this gives an exponential improvement in dependence on 1/εcompared to the sublinear algorithms of Goldreich, Goldwasser and Ron (JACM'98) and a linear speedup in n compared to the algorithms of Mathieu and Schudy (SODA'08). For the maximization version of k-Correlation Clustering problem our running time is O(k^4 n / ε^2) + k^O(1/ε^2), improving the previously best n k^{O(1/ε^3 log k/ε) by Guruswami and Giotis (SODA'06).