The Labour MP said there had been five recent Covid deaths in York, with four people in intensive care and more than 260 in hospital.
But the idea was knocked back by Health Minister Maggie Throup, who insisted the nation was "one step ahead of the virus" despite rising case rates.
Free LFTs will be scaled back in England from Friday, April 1, with only the over-75s and over-12s with weakened immune systems having access to them and the rest of the population being required to purchase them.
But in Wales, free lateral flow tests will continue to be available with symptoms until the end of June.
During a debate in Parliament on Monday night, Ms Maskell highlighted the concerning current situation with Covid in York.
She said: "In York, the case rate is now 977.7 per 100,000, 261 patients are in hospital poorly, five more deaths have just been announced and four people are in intensive care, so the virus is far from leaving us. In Labour-run Wales, an extension to the lateral flow test programme has been announced so that we can know where the virus is, manage it and protect our NHS. Should we not be doing that in England?"
She added it was also "vital" that people with relatives in care homes had access to tests.
"People have made huge sacrifices over the last two years in not seeing their loved ones in care homes, and not being able to afford a test will put another barrier in their way."
Health and Social Care Minister Maggie Throup said that while testing "has been a crucial countermeasure throughout the pandemic", the country is now in a "much better position" than in its initial stages.
She said: "We have severely weakened the link between infection and severe disease. As the Hon. Member for York Central said, cases are rising again. However, we are on the front foot thanks to vaccines and community Covid-19 treatments.
"The UK’s investment in groundbreaking vaccine technology and our world-beating vaccination programme has put us one step ahead of the virus."
She added that spring booster jabs are being offered to those aged 75 and over, residents in care homes for older adults, and individuals of 12 years and over who have a weakened immune system. Ms Throup said there is now also "widespread availability of targeted community Covid-19 treatments".
But Ms Maskell responded: "My concern is that in York, where about 90 per cent of people are vaccinated, the rate of people getting very poorly with covid is going up.
"The antiviral treatments are not effective, because there is an increase in mortality as well. Putting the additional line of defence—testing to prevent transmission—in the system is one way to save lives. I cannot understand why the Government will not move the issue on for three months; we could then review the situation again."
Ms Throup responded: "It is important that we should recognise that we have moved on. We have broken the chain of transmission with the vaccination programme, which is our first line of defence along with antivirals and therapeutics within NHS settings.
"Core to continuing to stay ahead of the virus and learning to live with Covid is a move to everyone embodying safer behaviours in their day-to-day lives.
"The UK Health Security Agency continues to monitor the virus and has recommended a package of contingency capabilities that form a reasonable insurance scenario to enable us to respond to resurgences or new variants of concern. I reassure the House that in line with recommendations, the Government have secured a supply of lateral flow devices to use if necessary. Limited ongoing free testing will be available for a small number of high-risk groups."