I just wanted to know if TXB0108 can work as level shifter Between Mega 2560 & 3.3V ST7565R( in parallel mode). Can anyone advice on this.
Should work, but looks a little bit overpowered. For parallel mode you will need two of the TXB0108 converters (parallel mode requires 11 or 12 lines), so you will waste 4 bit at the end. Moreover the bidirectional feature of the TXB0108 is not required (at least for u8g2 library). So, the 74HC4050 might be a more cheaper option.
BTW I would suggest to use the ST7565 in SPI mode if possible. Speed is almost the same but wiring is more simpler. Also only one 74HC4050 is required.
Oliver