In Sunday's vote, broadcast live on state radio, 183 out of 197 lawmakers present favoured the bill. The legislation must be approved by Iran's constitutional watchdog, the conservative 12-man Guardian Council.
Iran faces referral to the world body for possible sanctions after failing to convince the world that its atomic scientists are focusing on power stations rather than warheads.
The board of governors of the International Atomic Energy Agency (IAEA) meets in Vienna on Thursday to decide what steps to take with Iran's case. The bill is intended to give leverage to Iran's negotiators.
Lawmaker Alaeddin Boroujerdi, head of parliament's foreign policy and security commission, urged opposition parliamentarians from the reformist camp to show a united front in the national interest.
"This is not a factional, political issue - it is a national issue," he said in the debate.
The bill calls for Iran's government to stop following the Additional Protocol to the nuclear Non-Proliferation Treaty, which allows snap UN checks of atomic sites.
Top negotiator Ali Larijani threat-
ened to resume enrichment (file)
It calls on Iran to resume all nuclear activities that it stopped voluntarily. Foremost among these is the moratorium on enriching uranium.
Iran's chief atomic negotiator, Ali Larijani, has previously threatened to end snap checks and resume enrichment if Tehran's case is sent to the Security Council.
But parliament's bill turns this threat into law that the government must follow.
Despite Western pressure to halt its nuclear activity, Iran this week said it had begun processing a new batch of uranium. Iran insists on developing its own nuclear fuel cycle to produce fuel for power stations.
Washington fears Iran will enrich uranium to a high, weapons-grade level, rather than the low level needed for power stations such as the one it is building with Russian help at the Gulf port of Bushehr.
Gholamreza Aghazadeh, the head of Iran's Atomic Energy Organisation, attended the parliament meeting and said the bill sent a clear message to the IAEA.
"If the board of governors transgresses people's rights, nations are entitled to preserve their rights," he said.