Final answer:
The direct primary is the Progressive Era political reform that allows voters to choose party candidates to run for elected public offices.
Step-by-step explanation:
The political reform from the Progressive Era that allows voters to choose party candidates to run for elected public offices is the direct primary. The direct primary is an election where voters cast ballots to select nominees for upcoming elections. This reform ensured that voters could directly pick candidates for office rather than relying on party bosses to make the decisions.