No. MODE doesn't do anything useful to the parallel port hardware.
N2CPU's ICOM-RAM application will be bit-banging the parallel port registers and, as there isn't a maximum limit for any of the RAM access timings, Windows 3.1 or 9x won't interfere with it. Back in 2001, I was running a parallel port PIC and I2C EEPROM programmer based on David Tait's PP design, with my own DOS software written in Borland Turbo C (for MSDOS), and a quick look at the source code doesn't show anything special to add support for running under the Windows 98 SE I was using at that time, so I am *absolutely* certain there were no significant issues doing direct parallel port access from a DOS application under Windows before NT/2000/XP etc. However, you did have to make sure that none of the Windows printer drivers were attached to the port in question.
The most likely cause of the above failure is the port hardware doesn't support the bidirectional mode N2CBU's software expects to use, or possibly the port address is incorrect or not valid for MSDOS mode.
Without N2CBU's sourcecode, its fairly hopeless, which is why I proposed writing new software to use an Arduino.