Most research on trends in socio-economic fertility differences has focused on cohort total fertility and on women. This study aimed to analyse how cohort trends in parity-specific fertility differ across educational segments for men and women and what role multi-partner fertility plays in these trends. We used Finnish and Swedish register data on cohorts born in 1940-73/78. The main analyses used parity progression ratios, comparing ordinary ratios with similar ratios using births to first reproductive partners only. Among the low and medium educated, we observe strengthening parity polarization across cohorts, with increases in both childlessness and births of order three or higher, the latter largely reflecting increases in multi-partner fertility. Highly educated men and women more often have exactly two children. We demonstrate that cohort total fertility can mask significant parity-specific trends across educational groups and that changes in multi-partner fertility can play a part in cohort trends in socio-economic fertility differentials.