I second the mpv-based solution, as suggested by st1ck. If you prefer vlc, it now (as of the beta 4.0.0)[0] also supports dual/secondary subtitles. Making it work seems a bit fiddly: first you need to turn them on (under Tools > Preferences > Subtitles/OSD > Dual Subtitles (at the very bottom) > Align — change to anything but unset, and possibly also adjust offset). Then, when playing a video, to select them for that video, you need to "Toggle secondary subtitle control" with Ctrl+Shift+V (this means that the normal subtitle control shortcuts like "v", "Alt+v" etc. now apply to the secondary subtitles) and press "v" the right number of times to switch to the subtitles that you want.

(Obviously vlc and mpv will only work for DRM-free videos, e.g. from youtube or a DVD.)

If you were asking specifically about Netflix subtitles, there used to be an open source NflxMultiSubs extension[1] for both Firefox and Chromium, but it was broken by Netflix introducing changes to its video player and discontinued. There is an active, open-source dual-captions[2] extension, but it's for Chrome only. (However, since it's open source adapting it it for Firefox should be straightforward.) Finally, as an alternative approach, you could try a Firefox addon which allows loading arbitrary subtitles in the SRT format to netflix,[3] and which might perhaps allow you to have both netflix's subtitles and your own SRT ones at the same time.

[0] https://github.com/videolan/vlc/blob/master/NEWS#L30

[1] https://github.com/dannvix/NflxMultiSubs/

[2] https://github.com/mikesteele/dual-captions

[3] https://addons.mozilla.org/en-US/firefox/addon/netflix-srt-s...