Renaming and moving files in a subdirectory with an index and the folder name
Quite nicely done! I only have a few minor comments.

The script is not very "safe" as it is.
There are many possible failure points which are ignored:


What if /path/to/parent/directory/* doesn't exist?
What if cd "$dir" fails? ($dir is not really a directory, or you don't have permission to enter it)
What if cd video fails? (Like $dir)
What if *.mp4 doesn't exist?
What if mkdir "${newfolder}" fails? (no permission)


If any of the above happens, 
the script will continue to run regardless of any errors,
and potentially create junk files and directories you didn't intend.

Here's a safer way to protect against these failures:

for dir in /path/to/parent/directory/*; do
    cd "$dir" || continue
    result=${PWD##*/} 
    cd video || continue
    echo "$result"
    a=1
    for i in *.mp4; do
        test -f "$i" || continue
        dafile="$(printf "_%02d.mp4" ${a})" 
        dafolder="$(printf "_%02d" ${a})"
        newfolder="$result$dafolder"
        newfile="$result$dafile"
        mkdir "${newfolder}" || continue
        mv "${i}" "${newfolder}/${newfile}"
        cd "${newfolder}"
        ffmpeg -i "${newfile}" -r 30 -f image2 "${newfolder}-\%04d.png"
        cd ..
        ((a++))
    done
done


I also fixed the indentation, and replaced let a=a+1 with the somewhat simpler and more intuitive ((a++)).

You seem to like ${name} instead of $name. That's fine, but it's not needed anywhere in the posted code. Usually it's needed in situations like ${name}_suffix, where without the ${...} the "_suffix" would become part of the variable name.