Your ram could be an issue.
I had HUGE performance issues when I upgraded to parallels 9 and mavericks. I had to up my VM ram to 4 gb (which you only have 4) in order to get mine running smoothly.
So yes, I did get Parallels 9 and Mavericks to work fine. Why you cannot, I am not sure.
That is good that VirtualBox is working for you.
I would reinstall Parallels if I were you and tweak the settings. Make sure Parallels Tools was installed.
Construct 2 is not playing well with macs running Parallels 9* on Maverics**. Construct will open to the main page, but if you attempt to load a project or create a new project it will crash. Other applications have been having similar problems with parallels running on Maverics. The parallels team has said that they are working on fixes to alleviate the compatability issues with Maverics.
Is anyone running other VM software on Maverics with success?
*my version 9.0.24172
**My version OSX 10.9.1
13" MacBook 2008
2.4 GHz Intel Core 2 Duo
4 GB 1067 MHz DDR3
NVIDIA GeForce 9400M 256 MB