A survey's revealed almost half were worried about their children's wellbeing or education.
Parents and carers reported an increase in symptoms of stress, anxiety, and depression, especially during the period from November to December.
Symptoms included difficulty relaxing, being easily upset or agitated, feeling hopeless, and lacking interest and pleasure, feeling fearful and worried, as well as being more irritable, over-reactive and impatient.
More than 12,500 parents have now taken part in the Supporting Parents, Adolescents, and Children in Epidemics (Co-SPACE) study led by the University of Oxford.
Cathy Creswell, Professor of Clinical Developmental Psychology at Oxford and co-lead of the study said: "These findings build on others that suggested that parents were particularly vulnerable to distress during lockdown 1.
"We are particularly concerned about the level of strain felt by parents in low income families, those in single parent families, and those supporting children with special educational needs."
Indeed, the data show that parents and carers from certain households have been particularly vulnerable to elevated mental health symptoms.
Higher levels of stress, depression, and anxiety were reported by parents from single adult homes and lower income families (< £16,000 p.a.), as well as those who have children with special education needs and/or neurodevelopmental differences.
John Jolly, the CEO of Parentkind said: "Given the disruption to family life, it is vital that policymakers consult and listen to the concerns of parents on issues that directly impact them and their children's futures.
"Policymakers must give urgent consideration into how additional support for families most in need can be provided, before the disadvantage gap grows wide enough to create a lost generation."
Notably, parents who had young children (10 or younger) living in the household reported particularly high stress during the first lockdown and around a third of them (36%) were substantially worried about their children's behaviour at that time.
In contrast, a quarter (28%) of parents or carers who had older children only (11 or older) were worried about their children's behaviour during the first lockdown, yet nearly half (45%) of this group were worrying about their children's future.
A parent from the Co-SPACE study, Leticea Holland, said: "I think that UK leaders should have access to this data to see what is going on with the mental health of families and how they are being affected by Covid-19.
"I am also worried that the next three months will show a sharper increase in anxiety and stress where parents are having to do more teaching at home.
"Children are more worried as their teachers are becoming ill -and the 'new variant' sounds more scary, my daughter keeps commenting on an increasing worry of catching Covid-19 which she didn't do so much before."
Madiha Sajid, another parent taking part in the Co-SPACE study, said: ''Current times are hard enough as they are. As a working parent, the most important thing for me is to ensure my family's wellbeing, their safety, and their continued development.
"The Co-SPACE study has looked at several factors in depth; and personally, I have found it very useful to understand the different aspects of health and wellbeing for myself, and my child.
"Prolonged screen time, disruption to daily routine, frequent arguments, lack of exercise, and stress of exams have all been contributing factors to our mental health and wellbeing. The report will play an important part in informing policy, and developing interventions, which will hopefully benefit families all around the country.''
This research is tracking children and young people's mental health throughout the COVID-19 crisis. Survey results are helping researchers identify what protects children and young people from deteriorating mental health, over time, and at particular stress points, and how this may vary according to child and family characteristics.
This will help to identify what advice, support and help parents would find most useful. Crucially, the study is continuing to collect data in order to determine how these needs change as the pandemic progresses.