In a letter addressed to President Ebrahim Raisi and read at parliament, 250 MPs out of the 290-strong assembly wanted guarantees from the West that they would not withdraw from a new deal.
The vote came after the United States imposed fresh sanctions against Iran in July, targeting its missile programme. Parliament speaker Ali Larijani said "the Americans should know that this was our first action".
During the funeral processions, charged up Iranians joined their supreme leader in accusing regional rival Saudi Arabia of involvement in the twin bombings.
President Trump said the US would "grieve and pray" for the victims, but added states that "sponsor terrorism risk falling victim to the evil they promote."
Daesh claimed responsibility for the attacks on parliament and Ayatollah Khomeini's tomb in the Iranian capital. Iranian officials say a third attack was thwarted by the anti-terrorism department.
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