Your ssh session needs a controlling terminalneeds a controlling terminal, for that use ssh -t
Also, you can execute part2 from ssh directly. So part1 will look like this:
**SCRIPT1**
#!/bin/bash
target_server="1.2.3.4"
target_script="/home/user/script2.sh"
echo "Script 1 Started"
scp $target_script $target_server:/tmp/my_script_part2.sh
ssh -t $target_server /tmp/my_script_part2.sh
ssh $target_server rm -f /tmp/my_script_part2.sh
echo "Script 1 Ended"