The success of modern parallel paradigms such as MapReduce, Hadoop, or Spark, has attracted a significant attention to the Massively Parallel Computation (MPC) model over the past few years, especially on graph problems. In this work, we consider symmetry breaking problems of maximal independent set (MIS) and maximal matching (MM), which are among the most intensively studied problems in distributed/parallel computing, in MPC. These problems are known to admit efficient MPC algorithms if the space per machine is near-linear in $n$, the number of vertices in the graph. This space requirement however, as observed in the literature, is often significantly larger than we can afford; especially when the input graph is sparse. In a sharp contrast, in the truly sublinear regime of $n^{1-Ω(1)}$ space per machine, all the known algorithms take $\log^{Ω(1)} n$ rounds which is considered inefficient. Motivated by this shortcoming, we parametrize our algorithms by the arboricity $α$ of the input graph, which is a well-received measure of its sparsity. We show that both MIS and MM admit $O(\sqrt{\log α}\cdot\log\log α+ \log^2\log n)$ round algorithms using $O(n^ε)$ space per machine for any constant $ε\in (0, 1)$ and using $\widetilde{O}(m)$ total space. Therefore, for the wide range of sparse graphs with small arboricity---such as minor-free graphs, bounded-genus graphs or bounded treewidth graphs---we get an $O(\log^2 \log n)$ round algorithm which exponentially improves prior algorithms. By known reductions, our results also imply a $(1+ε)$-approximation of maximum cardinality matching, a $(2+ε)$-approximation of maximum weighted matching, and a 2-approximation of minimum vertex cover with essentially the same round complexity and memory requirements.