Why is serial EEPROM preferred over parallel EEPROM?
It is very simple. Number of pins and cost of packaging. 

EEPROM devices are primarily used to store parametric data or characterization constants for a device. The typical scenario is to write very seldom and read typically once each time the host device boots up. For this type of application the relatively slow writing times of EEPROM are of little concern. And the reading time to load at most a few K-bytes of data from a serial device (SPI or I2C) is not normally an excessive time impact. 

There is another factor that has played into the popularity of serial devices over parallel devices. That has been the migration of MCU devices from older microprocessor units with parallel busses to the much more prevalent modern types that have all their program storage memory and data memory built right on the chip. Often there is no longer a parallel bus option directly available. And in most applications there is very little interest in using up scads of pins to bit bang to a parallel peripheral.