So here are some quick research links.
Are parents becoming older?
yes in the Uk according to the ONS. The average age of mothers and fathers increased for the 10th consecutive year in 2018. The standardised mean age of both mothers and fathers has increased by a total of 4.2 years since they were at their lowest on record in 1975 and 1974 respectively
For a comparison look at EU fertility data to see the average age of becoming a mother in other European nations
The OECD also has a dataset for member nations on the average age of mothers
or the wider UN data on demography and age of mothers
Where is the best place to be a parent?
Look at the country rankings developed by BAV Group and The Wharton School of the University of Pennsylvania,
Save the Children Fund has produced regular reports and rankings relating to motherhood which use another set of indicators
A number of statistics map social protection/ benefits and allowances according to families
Useful to consult is the ILO Social Protection Report
Other categories to look at include Paternity/ Maternity rights at work. These are documented by the ILO. Look at this report on legal protections across the world 1993-2014 to trace trends. They also have more recent news and papers on maternity protection
Eurostat has expenditure on family and child benefits for European nations
and OECD trends in public expenditure on family benefits