I've got some Quick Parts that I don't use anymore but I'm not able to delete them. The option "Organize and Delete" is grayed out.
How am I supposed to delete Quick Parts?
It sounds like you are composing an email in Plain Text format then.
For some strange reason, the "Organize and Delete..." option is not available when composing in Plain Text. Funny enough, you are still able to add them though.
Select the Options (Outlook 2007) or the Format Text (Outlook 2010 and Outlook 2013) tab on the Ribbon to set the message format to HTML. Switch back to the Insert tab and "Organize and Delete..." will be available now.
For more about using Quick Parts see; Adding, modifying and deleting Quick Parts
For some reason, you are not allowed to organize or delete your
Quick Parts when in Plain Text mode.