This paper proposes a multi-objective visionbased navigation network for biomimetic underwater robots to cope with scientific observation, target selection, and obstacle avoidance in marine missions. Structurally, a stereo block attention module is first constructed to serially extract the channel and spatial attention portion of the real-time visual feedback. Next, the parallax attention mechanism is introduced to enable the network to excavate implicit parallax information in stereo pairs, effectively eliminating the oscillation of the network output in the presence of ambiguous visual input. Further, with the assistance of other low-cost sensors, the proposed navigation network can be expanded in some largescale application scenarios, such as sparse coral observation. Finally, underwater simulations reveal that the proposed method obtains significantly improved control effect and real-time ability, compared with other related works. In particular, based on a self-developed biomimetic robotic dolphin, collision-free simulations with a cumulative distance beyond 1000 m were carried out and validated the effectiveness and the superiority of the navigation network, where both dense and sparse targets were fully tested. The robotic dolphin can not only successfully conduct accurate coral observation without collision, but also quest the observation targets as much as possible in the area where the observation targets are concentrated. The proposed network provides an intelligent and efficient navigation scheme for autonomous underwater operation of small-size underwater robots.