Workaround for outputting HDMI audio from PC to Onkyo TX-SR506 receiver
You can get the audio out of your stereo but you cannot avoid at least one non-HDMI audio connection.

Option 1: Follow the method in the OEM Manual
Along with each HDMI input install a digital or analog audio input. Analog inputs are automatically associated with a device (DVD, DVR) but digital inputs must be associated with a device (p. 42 in manual).

This is the preferred method but requirs the most wires. Also your input device must have alternative audio outputs.

Option 2: Route audio back from the TV

Many TVs have a digital or analog speaker output. Use this as an input on your receiver. The catch here is you may not have a good way to select this input when you want it. On older receivers there was a Tape Monitor loop that would let you listent to Tape2 input while otherwise routing some other input out. It's not clear to me that your receiver is setup to do something like this. So then your TV out would have to be tied to a single HDMI input and you lose the capability of routing audio. Also you may find that there is a perceptable delay in the audio due processing by the Receiver + TV (but hopefully this won't be the case).