This study provides a systematic analysis of parity components of the cohort fertility decline in 32 low-fertility countries in Europe, North America, Australia and East Asia. We decompose the change in cohort fertility in each country among women born between 1940 and 1970 using parity progression ratios (PPR) derived mostly from census, register data, and large-scale surveys. We are also interested in how the effect of the changing parity progression ratios varied across broader regions and whether it was different in countries reaching very low completed fertility. The fall in fertility was mostly driven by reductions in the progression ratios to third and higher-order births in the first stage of 1940 to 1955 cohorts. Among women born between 1955 and 1970 fertility decline slowed down in most regions. This second stage of the cohort fertility decline analysed here is also distinguished by its regional diversity. The main distinction can be drawn between Central and Eastern Europe, where fertility decline was driven primarily by falling second birth rates and the German-speaking countries, Southern Europe, and East Asia, where the fall in first birth rates (and the corresponding rise in childlessness) had a stronger influence.