An AV Receiver has several types of inputs for audio and video, including several HDMI inputs, digital coaxial and toslink digital inputs, along with many 2-channel RCA audio inputs. This can include old style composite and component video inputs. An AV Receiver has multi-channel amplifiers for anywhere from 7 to 13 channels (you can hook up speakers directly to the receiver. The AV Receiver will also have RCA preamp outputs so you can hook it to any amplifier as well.
An HT Processor is pretty much the same thing, except it does not have internal amplifiers. It is a preamp/processor only and requires an external amplifier for all speakers. This allows the company to use more of the space inside for better power supply and analog circuits. Many HT Processors also have XLR balanced outputs.
For the best sound quality situation in your budget, I would recommend getting a used Marantz AV7702 processor. The 7702 is just recently been discontinued by Marantz and they can be found on ebay for anywhere between $800-1200 (or more if new-in-box).
If you don’t like getting used stuff, you can look for a new Marantz SR5011 or SR6011 AV Receiver. They will not have as good sound quality as the AV7702 processor. On the receivers, I generally recommend buying the highest one you can afford. The Marantz receiver models are pretty much functionally all the same. The difference is that the higher models have a better/larger power supply which translates into better sound quality. SR5011 is typically $799. SR6011 can be gotten from BH Photo & Video for $1149.
You can still use your Parasound amplifier with the receiver for main movie speakers. You can use a couple of the speaker connections on the receiver for the Zone 2 kitchen speakers (so you wouldn’t have to buy an additional amp).
I recommend connecting all audio/video sources (bluray, satellite, cd, etc.) to the processor/receiver and letting the processor/receiver decode and manage the audio/video. It will also have the benefit of doing bass management for your powered subwoofer. The Marantz devices also have Ethernet input so you can stream from DLNA or internet. If you want Netflix, you can hook up a Roku or use a Sony Bluray player to stream from Netflix