Puerto Rico's three political parties united yesterday in an effort to pressure Congress for fast action on legislation that could resolve the island's political future.

In letters to leaders of the House and Senate, the parties appealed for a compromise on the sharp differences between the two chambers over legislation that would authorize a referendum on the island's government.

The island's voters should be allowed to decide next year whether Puerto Rico becomes a state, remains a commonwealth of the United States or becomes an independent nation, the party leaders said.

Legislation authorizing such a vote passed the House on a voice vote in the final days of the last Congress, but died in the Senate. Although Sen. J. Bennett Johnston (D-La.), chairman of the Senate subcommittee that oversees territorial issues, has promised quick action on the issue next year, the island's political leaders have been skeptical, saying the House and Senate need to resolve their differences first.

The House legislation would require Congress to pass a second measure implementing whatever option the Puerto Rican voters select.

The Senate bill, drafted by Johnston, would adopt the voters' choice automatically without further action by Congress.

Earlier this month the Popular Democratic Party, Puerto Rico's largest political party, announced that unless House and Senate leaders agree on terms of a referendum by Feb. 19, it would not support any referendum legislation until 1993 on grounds the issue would confuse local elections set for 1992.

Yesterday, the governor's office in San Juan announced that the demand for quick action had been endorsed by the New Progressive Party, which favors statehood, and the Puerto Rican Independence Party.

"Otherwise we're going to have the same old movie," said Jose Berrocal, a spokesman for Gov. Rafael Hernandez Colon.

Without support of the island's political parties, members of Congress have said the referendum proposal is doomed.