Appearing before the education select committee, Spielman told MPs that new powers are necessary as children at illegal settings are at risk of radicalisation and abuse.
She said she wants inspectors to be able to seize evidence and prevent school staff removing documents and data, both of which they are currently unable to do.
Spielman previously called for new legal powers to tackle unregistered schools in December last year during the launch of Ofsted's 2017 annual report.
However, last month education minister Lord Agnew of Oulton said that the government had no plans to strengthen the inspectorate's powers.
"I would very much like to have stronger powers to get entry into suspected unregistered schools and to seize evidence," Spielman told education committee members.
"At the moment we can copy, we do take photographs where we can. But if the proprietors or teachers pick up everything that's there and walk out with it there there's nothing we can do. They can simply take it away."
Tougher checks are particularly needed in "authoritarian" religious settings where there is a greater risk that any incidents of abuse will be kept secret, said Spielman.
"We have a great deal of evidence that things that can go wrong for children in highly authoritarian structures that are largely shielded from public view," she said.
"That combination of secrecy and authority can allow bad things to happen."
She said Ofsted also needs greater powers to investigate concerns of radicalisation in religious settings.
Citing the case of Umar Haque, the unqualified teacher who was found guilty earlier this month of attempting to radicalise children, including pupils at the private Lantern of Knowledge Islamic school in London, Spielman said: "The Lantern of Knowledge case for example has shown us very clearly that very undesirable things are happening.
"In fact, in that case the evidence suggests that completely unknown, even to the management of that particular madrassa, children were being shown films of beheadings. They were engaging in Isis practice and being threatened."
Spielman also raised concerns that girls at unregistered religious schools are being taught that it is acceptable for them to be beaten.
Committee member Trudi Harrison, Conservative MP for Copeland, asked Spielman: "Do you believe that parents are sending their daughters to schools where they would be taught that they have to accept sex from their husbands and that beatings are acceptable in a relationship?"
Spielman said: "Whether knowingly or unknowingly, some parents are and that is very concerning."
Council leaders have called for local authorities to also have greater powers to crack down on illegal schools.
In September 2016 the Local Government Association criticised current legislation that says they can only enter the premises of a suspected illegal school if they have concerns about a child's safety.
Last year Ofsted revealed that 92 suspected unregistered settings were visited between 1 April 2016 and end of March 2017, with 31 warning notices issued and 22 closed.
