You have to break this down and try to isolate the problem.
1) Disconnect all the program sources. 2) Connect one only to the Parasound. Make sure it works right. Then insert the switch box with that only one program source (the same one) and see if it makes a difference. 3) Now start trying combinations of two and then three and finally four sources.
See if you can discern where the difference occurs.
A second series of tests is to hook the thing up and select a program source and run some music. But while running the music put a disc in two , three or four sources and hit play. Before you do any of these turn the Parasound volume all the way down for speaker safety. Then after hitting play on both items and the timer displays show they are running bring the Parasound volume up. If you hear more than one program the thing is wired wrong. This is unlikely if the thing has only printed circuit boards but if there is point to point wiring it possible that some kind of error was made in building it or even as mentioned above a design flaw. If you hear more than one program then I would throw the thing away. Can you return it? Open a case with Amazon? If you hear more than one program each source is driving not only the input impedance of the Parasound but also the output stage of the other devices. This should not be but if it is it is likely the problem.
How long are the cables. High frequency loss due to too long cables is possible, but should be unlikely. The speakers in my avatar were driven by
by cables that ran from the living room to the power amps downstairs in the basement using RCA cables I made that were very long maybe 30 40 feet.
Of course I had shelving controls for the high frequency amps. If the unit passes the above test, try to get the shortest cables that will work. The capacitance of cabling is often a direct function of length. Capacitance in parallel with the output COULD roll off high frequencies (but like Sal I consider this rather unlikely.)
See if you can find out 1) the input impedance of the Parasound. Being solid state it may be very low. (ie 500 or 600 ohms) and 2) the output impedance of your various program sources which if they don't have enough buffer amplification might not be able to drive the Parasound if it is too low. This all should be unlikely but you may have equipment that just is a little bit incompatible.
Are you only using one of the two outputs the unit has? Does it do the same thing when you use both outputs? The one I sent you the link to only has one output which would probably be a better way to configure all this. (Edit: just reread how you are using this.)
It looks like the way this opens is that four screws on the front and four on the back allow the respective panels to come off. One panel probably has a circuit board and the other some wiring from the panel do said board. So becareful taking them apart so you don't pull the one with wiring apart. It hopefully is soldered and there should be enough slack to get a look.
You can buy a volt ohm milliameter at Harbor Fright for less than ten dollars and with this you could do some probing to see if there is conductivity where it does not belong. Amazon too for even less money.
https://www.harborfreight.com/7-function-digital-multimeter-63759.html
You can get a sound test CD that will turn you players into a test generator.
https://audiophilestyle.com/ca/ca-academy/drzs-test-cd-free-download-r780/There are many others. Notably from Stereophile , Denon that can be found.
Some of these can use your sound card as Sal suggested as a measuring tool. There are lots of such tools including Adobe Audition which is out there in a version for free. Audacity which is free and many others.
You can also get a radio shack style sound level meter to measure the output and it may have enough high frequency response to tell if there is a roll off.
https://pages.uoregon.edu/baker/tools/toollibrary/RS (Sound Level Meter).pdf