The children required wheelchair assistance on the July 2025 flight because of a condition that affects their mobility, balance and endurance.
They were allegedly discriminated against as the plane flew from Brisbane to Denpasar in Bali.
The western Sydney family are representing themselves in a Federal Court lawsuit launched against Virgin Australia in February.
They claim cabin crew denied both children access to the forward aircraft lavatory publicly in the presence of other passengers.
"The (twins) were instead directed to traverse the full length of the aircraft cabin without assistance, notwithstanding (Virgin Australia's) prior knowledge of their reduced mobility," court documents seen by AAP on Tuesday say.
No reasonable adjustments were offered, making the travel environment inaccessible and unsafe for the children, the parents wrote.
As a result, the twins experienced significant psychological distress and functional regression, including heightened anxiety, fear associated with air travel, and increased dependency on their parents, the pleadings say.
They have also sued the Department of Foreign Affairs and Trade over alleged failures to provide assistance after they became stranded in Bali in August 2025.
The government compounded the harm by offering inaccurate information to other commonwealth agencies and prematurely withdrawing support, the pleadings say.Â
Virgin Airline's parent company Bain Capital allegedly failed to intervene in August 2025 to prevent further discriminatory conduct.
"(This) conduct disrupted the family's support arrangements, including reliance on extended family carers, and caused ongoing anxiety and loss of confidence in accessing public and transport services," the parents wrote.
The mother and father claim they experienced psychological distress and trauma from seeing their children discriminated against and having to deal with being stranded.
The family is seeking compensation including damages and aggravated damages from Virgin and Bain, and orders requiring the airline to correct its governance and policies.
Before reaching the Federal Court, the matter went before the Australian Human Rights Commission where it made a decision to terminate the complaint in December 2025.
Delegate Rachel Holt found the discrimination claims against Virgin were reasonably arguable but could not be settled through private negotiations.
She found the claims against Bain and DFAT lacked substance.
Bain was not aware the children had disabilities, Ms Holt said.
Furthermore, any alleged shortcomings in the federal government's consular assistance were not related to the boys' physical ailments, she found.
In submissions to the human rights commission, DFAT denied breaching discrimination law, saying the family could not say how they had been treated less favourably than someone without a disability.
A Virgin Australia spokesman declined to comment on the ongoing court case apart to say the firm was aware of the allegations.
DFAT has been contacted for comment.
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