Satellites and civil aircrafts (CAs) with computing ability are valuable access platforms, making it possible for Internet of Things (IoT) devices to offload their computation-intensive tasks in remote areas without network infrastructures. Unlike existing works mainly focused on the static scenarios or the interaction between any two types of local, edge and cloud nodes, we propose an innovative multi-tier hybrid parallel computation architecture in CA-augmented space-air-ground integrated networks (CAA-SAGIN). Specifically, devices perform local computing, CAs and satellites act as edge servers, and ground stations of satellite networks operate cloud computing. Aiming to minimize the weighted sum of end-to-end (E2E) delay and energy consumption, we formulate a partial computation offloading problem by jointly considering access strategy, transmit power, computing resource allocation, offloading ratio and delay tolerance. The platform selection exists both within and between layers, and there are inner- and inter-coupling relationships between communication and computing resources. The issue is solved by the proposed multi-tier partial task offloading (MPTO) algorithm. The original problem is firstly decomposed into primal and master subproblems by generalized benders decomposition (GBD) method, and parallel successive convex approximation (SCA) theory is utilized to transform the multi-variable NP-hard master problem into a convex one. Simulation results demonstrate the convergence and optimality of the MPTO algorithm and the advantages of this multi-tier hybrid computation offloading system. Also, the optimal tradeoff between E2E delay and energy consumption can be achieved by the MPTO algorithm.