This is a problem for me too. My Quicken can look like Agmanbrad's photo, except if one were to have a register contents displayed (I understand there would be privacy issues) the problem would be revealed to be much worse -- text so big on the register lines that the software cannot be used. I have several points to make.
1. This problem also occurs for me with Microsoft Outlook. Like Quicken, Outlook has multiple sub-windows within its main window that use different fonts and different font sizes.
2. For me, this problem did not occur at first use. It first happened when I switched from using my MBP at my desk, where an external 1920x1200 monitor is connected, to portable MBP use, where I use the MBP's retina screen. As soon as I saw Quicken on the MBP display, its appearance had gone crazy with the double menu and wild font sizes. I have applied a Windows 8.1 registry tweak that makes the windows' borders very narrow. I noticed that when the Quicken and Outlook fonts/font sizes/menus went all crazy, that I had also lost the narrow-windows-borders registry tweak. So, when this glitch occurs, it also affects seemingly unrelated parameters in the Windows registry. I was able to fix it when it happened the first time by restoring a recent backup of my virtual machine file. Everything was back to normal, until it happened again on a subsequent display change. The various scaling workarounds suggested are just that -- workarounds, and do not restore the proper fonts and text sharpness to Quicken or Outlook.
3. Oddly, this issue doesn't occur every time I switch displays. Sometimes it does not occur. I think it's related to whether Quicken or Outlook are open, or are minimized or on screen. I haven't figured out if there is an exact combination, or if it just occurs randomly.
4. My workaround is to keep a recent VM file backup at the ready in another folder. When the problem happens, I close the running VM, overwrite it with the backup copy and start it up. Then I can continue working with proper Quicken and Outlook appearances. (I keep my data outside the VM file, so this procedure does not cause me to lose any data.)
5. This problem has been reported many times. A quick search revealed this thread and three others.
https://forum.parallels.com/threads/parallels-9-on-retina-mac-double-menus.295500/#post-764930
https://forum.parallels.com/threads/huge-problems-parallels-8-and-windows.262468/
https://forum.parallels.com/threads/parallels-9-and-quicken.292557/
6. I started using Parallels at Parallels 9. The problem has continued unchanged through Parallels 10 and now 11. I think it's really time for the Parallels team to properly fix this.
7. I am presently running a 30-day trial of VMware Fusion. I have deliberately tested for this problem on Fusion, and am happy to report that it does not occur. I can switch between my two displays endlessly with no font/resolution/double menu/etc. issues. Unfortunately Fusion has a few other usability issues. Parallels is clearly ahead in Mac-Windows integration. I'd like to stay with Parallels, but will move to Fusion if there's no coming solution to this.