How to Stop a Background Process When the Foreground Process Finished Completing the Task

Need to stop b.sh when a.sh is finished. Both scripts must run in parallel one must not wait for the other. b.sh contains the profiling of the process a.sh. When I run the script while loop does not stop. I couldn't use $! to kill the background process because other processes are running in the background created from a.sh script file.

Contents of a.sh:

#!/bin/bash 
pid=$$
export pid
./b.sh &
b_id=$!
#load time
docker load<ubuntu.tar
id=$(docker images -a | awk '{print $3}' | awk 'NR==2')
docker run -t --rm $id
echo "Container started"
pkill -P $b_id

Contents of b.sh:

#!/bin/bash
while true; do
some profiling code
done