I installed DIP sockets and put some different memory chips (i used chips of different manufacturers as I don't have a complete set of the same chips) - and that problem with Memtest still appears. Funny thing is that these "eeeeeeeeeeeeee" errors turned green now. Maybe I still have some defective chips there?
I also tried Memtest 4.0 and it just halts with unexpected interrupt at the beginning. Another funny thing is that this computer starts even with all memory modules removed, and with no onboard memory - and it shows First 64KB memory Error or something like that (and it halts there).

Does anyone have some other DOS utilities for testing stability of 386SX?

did you try different versions of memtest ?

Yes and I couldn't get it to work, but finally I tried GoldMemory and it works (very slow), it never has shown any problems. But the parity problems still were showing up sometimes, usually after the system is in idle, and I press a key. I soldered DIP sockets, but I don't have full set of new memory chips, I ordered some and will do some testing later. Anyway, I haven't seen a parity check error for a while, but I have another problem.

I installed a sound card and some small speakers. The card is ES1688F, but the same issue is also with Crystal CS4235. When I start Prince of Persia (1.3), with sound blaster enabled in setup, computer freezes at Loading... screen, HDD led is red for a minute or two, then a message Drive C: is not ready, try again (it's a game message, not system message)... or drive A: when I run it from floppy disk, shows up. Game works fine when I disable digital sound and use only AdLib mode. But Prehistorik 2 works with digital sound without problems. I also installed Windows 95, just to do some testing, and when I run Prince of Persia in Windows, both digital sound and music work fine (tried that with ESS). I'm using only the default sound drivers. Does anyone know how to make the sound work in pure DOS?

I also would try another Powersupply (new) this can cause this sort of problems too.
I also had the same Modell which had a few bad capacitors (capacity loss) inside the power supply

All right, the investigation goes further...

I disabled the HDD in BIOS and booted PC from floppy disk. I set the BLASTER variable manually and ran Prince of Persia from floppy disk. Sound and music works fine.
So, I guess that the HDD controller is somehow interfering with the sound card. Unfortunately i can only disable or enable HDD, there aren't may options in BIOS menu.

The ES1688F sound card has built-in IDE connector. I am also using 3com 3c509b network card with ROM socket. Do you think that those issues could be fixed by installing XTIDE BIOS into network card's socket? I never used XIDE bios. But I guess that it would give me some configuration options, or even allow to plug the HDD into sound card's IDE connector?
I think that I should also try using some dynamic drive overlay software, maybe it will help.

I also noticed that sometimes I was getting some heavy stuttering in digital audio when playing Prehistorik 2. Sometimes it was okay, sometimes not. I don't know how to reproduce it. But the problems probably appear only when sound card is used in 16-bit ISA slot, in 8-bit slot it's fine.

//update:
Sound in PoP seems to work when booting from floppy disk even with HDD enabled. BUT only when I run PoP from floppy disk, when I go to C: and run it from HDD i get drive not ready error. So it probably confirms that it is some kind of controller and sound card conflict. Any other opinions and ideas?

//update2:
Great, I managed to make a workaround - but I'm not 100% satisfied. I created a 1MB ramdrive, copied the game there and it works just fine. So we can clearly see that there is a problem with HDD controller. Unfortunately I couldn't install drive overlay software because the HDD is only 120MB so it wouldn't let me install. Are there any other options to fix that?

//update3:
Well, this batch script (prince.bat) + ramdrive works pretty well... the only downside is that it's wasting 1MB of RAM.

1deltree /y d:\ 
2md d:\pop
3xcopy c:\prince\*.* d:\pop\*.* >NUL
4cd d:\pop
5d:
6d:\pop\prince.exe
7xcopy d:\pop\*.sav c:\prince\*.sav /Y >NUL
8xcopy d:\pop\*.cfg c:\prince\*.cfg /Y >NUL
9c:
10deltree /y d:\ >NUL

DUDE YOUR TOSHIBA HAS A VIRUS!!! I just ran into this exact same problem on a Toshiba T3100e, found this thread, and was convinced it was some kind of hardware issue. Then I put a floppy disk into my modern PC to transfer files over, and Kaspersky notifies me that the A drive is infected with Virus.Boot.Parity.b. Looked it up and found this wiki page which explains everything:

Systems infected with the Parity Boot virus may experience the following message being displayed after keys are pressed on the s […]
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Systems infected with the Parity Boot virus may experience the following message being displayed after keys are pressed on the system keyboard:

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1PARITY CHECK

I know it's been five years since you posted this thread and you may no longer even have the computer in question, but I thought I'd let you know what I discovered! It's not a memory problem at all!