What OS do I install on parallels to run Quartus II and other design software?
1) It will probably work on windows 8 just fine.  It's more likely that you would have issues with device drivers for programming cables and the like than with regular software.  

2) Just install windows 7.  The windows 8 interface is rather whacky and it may be cumbersome to navigate the 'edges and corners' tablet paradigm when the OS is running in a window.  

3) I have no problems running Altera Quartus II and Xilinx ISE under Arch Linux; you should have little trouble with Ubuntu.