The post-quantum key encapsulation mechanism, CRYSTALS-Kyber, has recently been selected by the National Institute of Standards and Technology (NIST) for standardization. Consequently, it becomes crucial to assess the resistance of CRYSTALS-Kyber implementations to physical attacks. While side-channel attacks on embedded software platforms have been well studied, this work introduces a novel attack on hardware implementations of CRYSTALS-Kyber. Initially, we propose a multi-ciphertext message recovery attack that is capable of extracting messages from side-channel measurements of parallel message encoding. Building upon this, we further develop a key recovery attack based on an optimal ciphertext-choosing strategy that maximizes key recovery accuracy, as well as a lattice reduction attack capable of solving the entire secret key even when confronted with an imperfect side-channel distinguisher. To evaluate the effectiveness of our attack, we conducted experiments on a Xilinx FPGA board. Our results demonstrate that our attack is capable of successfully recovering the secret key using 96 power traces, with a success rate of $100 \%$. This study reveals that parallel implementations remain vulnerable to sidechannel attacks, underscoring the necessity of additional analysis and countermeasures for lattice-based schemes implemented in parallel.