How can I get my Macbook Pro with Yosemite OS and Parallels Desktop 10 to run a Windows only application?
I run Parallels in the environment you describe and that is not how Parallels behaves for me.  I believe your installation or configuration is corrupt / invalid somehow and you will definitely need to contact Parallels support and / or start over with your Mac OS / Parallels installation.  Personally I recommend contacting Parallels support.  

For myself, all Parallels drop down menus appear and work fine.  I am able to switch from Coherence mode to Full Screen mode, etc. easily without issue, and of course launch Windows only applications just fine.

So far as I can see, there is no answer to your question other than to reach out to Parallels support and / or start with a fresh installation.  Once setup Parallels works great by the way.  Good luck!