TOKYO—Political parties in Japan are campaigning for greater freedoms for women and putting record numbers of women on the ballot in a national election this month, seizing on the momentum of the country’s shifting economic landscape.

Prime Minister Shinzo Abe has made the idea of “womenomics” central to an economic-revival plan that seeks to mobilize untapped segments of the labor force to spur growth when the working-age population is in decline.

During...