It's not a browser-based solution, but, the mpv media player can display subtitles while playing an audio file.

If you run mpv --subfile=something.srt something.mp3 it will show the subtitles in the terminal, and to get them in a window instead you can add the --force-window commandline option.

You can probably download the audio (and maybe subtitles?) from whatever website using yt-dlp, or maybe even stream from the site using mpv directly by just giving it the URL.