They warn that such internships without a salary, used as stepping stones into jobs, are a financial block to those who cannot afford to work unpaid.
The All Party Parliamentary Group on social mobility says all interns should be paid after their first month.
Justin Madders, the group's chair, called for better access to "top jobs for those from less advantaged homes".
The "Class Ceiling" study follows concerns that jobs in areas such as law, medicine and the media are disproportionately filled by socially-advantaged, privately-educated youngsters who have attended top universities.
The cross-party social mobility report highlights the way that entry into some professions can be skewed in favour of more affluent youngsters.
The use of unpaid internships as an entry system means that it is only available to young people who can either support themselves or who have parents who can provide for them while they are working without pay.
There is also a geographical barrier when internships are in London, with such unpaid work much more feasible for young people from the capital who can live with their parents.
The report cites research showing that almost a third of graduates working as interns were not being paid.
But an attempt in Parliament to introduce such a ban on unpaid internships was blocked last November.
Tory MP Alec Shelbrooke had labelled them the "acceptable face of unpaid labour in modern Britain". But his bid to outlaw unpaid internships was opposed by the government.
As well as calling for a ban on unpaid internships, the All Party Parliamentary Group on social mobility wants work experience to be more widely available and "fair and transparent".
Work experience has become an important starting point for many jobs and there are concerns that this could freeze out youngsters without family connections in the professions.
The report says that all work experience posts should be "publicly advertised to allow a more diverse range of candidates to apply".
The report calls for employers to make better use of "contextual" information about applicants, such as looking at their results in terms of the type of school they attended and any disadvantage they might have overcome.
Mr Madders, MP for Ellesmere Port and Neston, said: "If the current government is serious about improving access to top jobs for those from less advantaged homes, they need to take a much more strategic approach.
"This means linking the work of schools, universities and employers to build a real business case and practical plan for improving social mobility."
Alan Milburn, chair of the Social Mobility Commission, said he backed the calls for an end to unpaid internships.
He said firms needed to "wake up and realise that it makes sound business sense to recruit people from all backgrounds".
Mr Milburn added: "Research has consistently shown that people from more affluent backgrounds, who attend private schools and elite universities, take a disproportionate number of the best jobs while those from poorer backgrounds are being systematically locked out."
Unpaid internships should be banned as a barrier to social mobility, says a report from MPs and peers.