Internships are never given by large companies or organisations on parental contacts.
DS applied for an internship with a major European bank at the end of his first year, which he did not get. A friend of DDs did get it despite being still at school when is was advertised for UGs. Her father was a very senior Director of the same bank. I do not believe she went through the same recruitment process.
At the same time a friend of DS' got an internship with a major US company which her dad happened to work for. I asked about the blind recruiting that was supposed to happen. Easy apparently. The dad just hand carried her application into the personnel department.
However the real advantage kids from middle class background have is that they understand the importance of internships. By the time they graduate many will already have a pretty full Linked in profile of different experiences. Which is not just the experience itself but also the experience of applying, the effort that is required, the experience of interviews and so on. My observation is also that these kids are more likely to have decided on a career and will then have decided what to study and where. If you know where you are going it is easier to get there.
In contrast others seem to chose a subject they enjoy, and to then decide to enjoy their University years, only starting job search after they graduate. They will probably be fine but won't be off to the same running start.
Connections can help, as my examples above illustrate, but we have also seen kids who have applied for literally hundreds of internships and jobs, and yes, they land one in the end. And you only need one. (This has just happened to a friend of DS who comes from a country with a despotic regime and who can't return, but who now has three or four really good offers. Ditto an au pair we knew who after four or more years volunteering each weekend finally got her foot in the door and who has now been promoted to a conservation/environment role that would be the envy of many environmental science majors.)