VQ-VAE, as a mainstream approach of speech tokenizer, has been troubled by ``index collapse'', where only a small number of codewords are activated in large codebooks. This work proposes product-quantized (PQ) VAE with more codebooks but fewer codewords to address this problem and build large-codebook speech tokenizers. It encodes speech features into multiple VQ subspaces and composes them into codewords in a larger codebook. Besides, to utilize each VQ subspace well, we also enhance PQ-VAE via a dual-decoding training strategy with the encoding and quantized sequences. The experimental results demonstrate that PQ-VAE addresses ``index collapse" effectively, especially for larger codebooks. The model with the proposed training strategy further improves codebook perplexity and reconstruction quality, outperforming other multi-codebook VQ approaches. Finally, PQ-VAE demonstrates its effectiveness in language-model-based TTS, supporting higher-quality speech generation with larger codebooks.