findmnt --raw --noheadings --output options --target SOME_DIRECTORY | grep -qE '(^|,)ro($|,)'

• findmnt is a handy alternative to mount when writing scripts
• SOME_DIRECTORY does not have to correspond to a mountpoint. If it doesn't, findmnt will traverse its parent directories until it finds the corresponding filesystem / mountpoint.
• a simple grep ro would also match options like errors=remount-ro, so we make sure to only match the single ro option. It must be delimited by commas or the start/end of the option string.