It looks as if Godaddy, like many SSL issuers, use an intermediate certificate that must be served by your SSL server in order for the chain of trust to be complete.
In essence, instead of signing your CSR with a certificate which is itself in the public bundle known to most browsers, they sign your CSR with a certificate of their own; this certificate of theirs is in turn itself certified by one of the certificates in the browsers' public bundle to sign anything.
There are good reasons to do this, but the upshot is that when you serve your certificate to people in the SSL handshake phase you have to serve a copy of godaddy's signing certificate at the same time. Then the browser can say to itself "the site certificate is signed by this intermediate godaddy certificate, and the intermediate godaddy certificate is signed as 'OK to sign other things' by eg Equifax/Thawte/Verisign/some other top-level authority whom I trust", and the browser is happy. If the browser doesn't get that intermediate certificate, it can't connect the chain of trust, and it isn't happy.
Godaddy have a chain certificate installation instruction page for apache at this help page.
Edit: it sounds like your SSL config has more than what you wrote. You can't just add config to apache and expect it to work, you have to remove anything that conflicts. Try
find /etc/http/conf -type f -exec grep -i sslcertificatefile {} /dev/null\;
(replacing /etc/http/conf with your apache config root, if you keep it elsewhere) and see where the plesk certificate is configured in. Commant that section out and try restarting apache.
Best Answer
To make apache receive the passphrase everytime it restarts, add this to the httpd.conf:
in your passphrase-file:
and make the passphrase-file executable: