The only way to allow the application on subdomain.example.com to read a cookie from www.example.com will be for www.example.com to set the example.com top-level cookie.
This would allow subdomain.example.com to read it, but it would also allow every other example.com subdomain to see this - that you said you did not want to.
: cookie . cookie , . , cookie example.com , - subdomain.example.com, BOTH example.com subdomain.example.com.
TL; DR
cookie , , cookie (, ). Google Analytics, .