Bash – Copy files in same folder and replace year in its name

To get the YEAR portion of the filename MY_AMAZING_FILE_THAT_IMADEIN_YEAR_TY.pdf without using external utilities:

name='MY_AMAZING_FILE_THAT_IMADEIN_YEAR_TY.pdf'

year=${name%_*}    # remove everything after the last '_'
year=${year##*_}   # remove everything up to the last (remaining) '_'

After update to the question:

Moving PDF files from under topdir to a directory /some/path/destination/<year>/<parent> where <year> is the year found in the filename of the file, and <parent> is the basename of the original directory that the file was found in:

find topdir -type f -name '*.pdf' -exec bash ./movefiles.sh {} +

movefiles.sh is a shell script in the current directory:

#!/bin/bash

destdir='/some/path/destination'

for name; do
    # get basename of directory
    parent=${name%/*}
    parent=${parent##*/}

    # get the year from the filename:
    #  - Pattern:  _YYYY_         (in the middle somewhere)
    #  - Pattern:  _YYYYMMDD.pdf  (at end)
    if [[ "$name" =~ _([0-9]{4})_ ]] ||
       [[ "$name" =~ _([0-9]{4})[0-9]{4}\.pdf$ ]]; then
        year="${BASH_REMATCH[1]}"
    else
        printf 'No year in filename "%s"\n' "$name" >&2
        continue
    fi

    # make destination directory if needed
    # (remove echo when you have tested this at least once)
    if [ ! -d "$destdir/$year/$parent" ]; then
        echo mkdir -p "$destdir/$year/$parent"
    fi

    # move file
    # (remove echo when you have tested this at least once)
    echo mv "$name" "$destdir/$year/$parent"
done