Cross-domain recommendation (CDR) has been attracting increasing attention of researchers for its ability to alleviate the data sparsity problem in recommender systems. However, the existing single-target or dual-target CDR methods often suffer from two drawbacks, the assumption of at least one rich domain and the heavy dependence on domain-invariant preference, which are impractical in real world where sparsity is ubiquitous and might degrade the user preference learning. To overcome these issues, we propose a Multi-Sparse-Domain Collaborative Recommendation (MSDCR) model for multi-target cross-domain recommendation. Unlike traditional CDR methods, MSDCR treats the multiple relevant domains as all sparse and can simultaneously improve the recommendation performance in each domain. We propose a Multi-Domain Separation Network (MDSN) and a Gated Aspect Preference Enhancement (GAPE) module for MSDCR to enhance a user's domain-specific aspect preferences in a domain by transferring the complementary aspect preferences in other domains, during which the uniqueness of the domain-specific preference can be preserved through the adversarial training offered by MDSN and the complementarity can be adaptively determined by GAPE. Meanwhile, we propose a Multi-Domain Adaptation Network (MDAN) for MSDCR to capture a user's domain-invariant aspect preference. With the integration of the enhanced domain-specific aspect preference and the domain-invariant aspect preference, MSDCR can reach a comprehensive understanding of a user's preference in each sparse domain. At last, the extensive experiments conducted on real datasets demonstrate the remarkable superiority of MSDCR over the state-of-the-art single-domain recommendation models and CDR models.