"Der zweite Mann"
Translation:The second man
You're right that it is nominative masculine, but the reason it doesn't end in -er is because the adjective follows the definite article Der. In nominative case, adjectives following a definite article end in -e except for plurals, which is -en.
So is it correct to write "Die zweiten Eltern" or "Die Dritten Kinder" or finally "Die ersten Männer"?