It can be hard to mesh older tech with new tech. I do it for a alot for people and myself. It can work out quite well if you have the equipment to do the mixing of the technology. HTisB are made to be cheap and give an entry level path for people to surround sound. There isn't a problem with that, it's just better to buy something better so you aren't stuck with something that will be worthless in a few years.
Kinda hard to say without what models you are using but just going to the red/white cables(left and right sound) will only produce a stereo sound(2.0)
Depending on if the Panasonic has some virtual sound format like Dolby Pro, that will take the stereo sound and virtualize the signal. If your TV has the red/white, you can try that but like I mentioned above, something will have to expand the sound to surrond.
Also, you may want to updrade your Dish DVR so it has HDMI which I'm probably sure your TV has, as a first step in upgrading your situation. I know that doesn't help or solve the HTiB problem. At least your upgrade factor will be good for a few more years! hehe.
Anyways get back to us with the models and if you have a budget if you want to upgrade.
Happy listening, the Prisoner...