Recently, the rapid development of the Internet of Things (IoT) has led to an increasing exponential growth of non-scalar data (e.g., images, videos). Local services are far from satisfying storage requirements, and the cloud computing fails to effectively support heterogeneous distributed IoT environments, such as wireless sensor network. To effectively provide smart privacy protection for video data storage, we take full advantage of three patterns (multi-access edge computing, cloudlets and fog computing) of edge computing to design the hierarchical edge computing architecture, and propose a low-complexity and high-secure scheme based on it. The video is divided into three parts and stored in completely different facilities. Specifically, the most significant bits of key frames are directly stored in local sensor devices while the least significant bits of key frames are encrypted and sent to the semi-trusted cloudlets. The non-key frame is compressed with the two-layer parallel compressive sensing and encrypted by the 2D logistic-skew tent map and then transmitted to the cloud. Simulation experiments and theoretical analysis demonstrate that our proposed scheme can not only provide smart privacy protection for big video data storage based on the hierarchical edge computing, but also avoid increasing additional computation burden and storage pressure.