How do I get osascript to parse backslashes and run this command in a new Terminal window?
Because it's going through another layer of interpretation, you need to escape the escapes -- essentially, you need to double the backslashes:

osascript -e 'tell app "Terminal" to do script "echo -n -e \\\\033]0\\;My Custom Window Title Here\\\\007 ; ping www.google.com"'