I don't believe that there is any way to access another web users email account, much less read their email. PHP can access IMAP servers if you know the email address and the password, and even then, normal email servers do not count the number of times and email has been read. I only know of being able to see if the email has been read via IMAP
One thing you could do, is create a little icon that you could put on the bottom of an email and then create the image source to be a PHP page, such as picture.php, where it would connect to the database and increment the count. You would have to use some query variables so you can accurately track which email is requesting the picture
If you are attempting to send them an actual PHP page with coding in the email, you will be unsuccessful, since PHP is a server-side language. Meaning, it is just like you renaming the PHP file you are sending as an email to HTML and then opening it. (ex picture.php => picture.html)