The only unqualified political party in New York that has more than a single candidate for any federal or state office (not counting fusion nominations) is the Medical Freedom Party. It placed one of its own nominees on the ballot for U.S. House, and four for legislature. It also nominated a few Republican nominees. Here is its webpage.

The Independence Party, another unqualified party, also completed some legislative petitions, but they are all instances when the Independence Party nominated the Republican nominee.

On October 3, three of the candidates on the ballot for U.S. Senate in Idaho debated each other. See this story. They were the Republican and Democratic nominees, and the independent candidate, Scott Cleveland. The Libertarian Party and Constitution Party nominees were not invited into the debate.

Brazil held an election on October 2. See the wikipedia article here. For president, eleven candidates were on the ballot. No one got a majority, so a runoff will be held October 30.

For the Chamber of Deputies, which has 513 members, twenty-three parties won at least one seat.

For the Senate, which has 108 members, fifteen parties won at least one seat.