Tool name: Exiftool
You can use this command to set the common EXIF date/times from your file names:
exiftool "-alldates<20$filename" DIR
This should work assuming that all years are 2000 or later. Because you only have 2 digits in your year, I have added “20” before the file name to complete the 4-digit year.
I think you can do it with ExifTool:
exiftool -v "-FileModifyDate>DateTimeOriginal" *
Tried on some copies of jpegs and it seems to have worked.