In the CYCLE PACKING problem, we are given an undirected graph G, a positive integer r, and the task is to check whether there exist r vertex-disjoint cycles. In this paper, we study CYCLE PACKING with respect to a structural parameter, namely, distance to proper interval graphs (indifference graphs). In particular, we show that CYCLE PACKING is fixed-parameter tractable (FPT) when parameterized by t, the size of a proper interval deletion set. For this purpose, we design an algorithm with O(2O(tlogt)nO(1))$\mathcal{O}(2^{\mathcal{O}(t\log t)}{n}^{\mathcal{O}(1)})$ running time. Several structural parameterizations for CYCLE PACKING have been studied in the literature and our FPT algorithm fills a gap in the ecology of such parameterizations. We combine color coding, greedy strategy and dynamic programming based on structural properties of proper interval graphs in a non-trivial fashion to obtain the FPT algorithm.