I think a better question might be: When were Italians no longer considered inferior to Protestant, British (or perhaps French)-born [edit: born/descended, and of course, American born] Americans?

There were certain scientific theories which were used to try to "racialize" Italians (and Jews, Irish, etc.). These included craniometry and phrenology, the study of skull and brain formation, respectively. I don't think anyone would have considered Italians to be on the same level of the racial hierarchy as blacks. Honestly, a lot of the discrimination Italians faced would have probably been based on their religion and spoken language more than their perceived "race." Class also. Ronald Bayor touches on these ideas in "Neighbors in Conflict," which assesses ethnic hierarchy in New York City in the late-nineteenth century. David Roediger's "Wages of Whiteness" would also be a good place to look. Though it focuses mainly on white on black racism, there are interesting diversions into the "shades" of ethnic hierarchy.

As to when Italians found more acceptance (in America, anyway) I would say the 1920s, generally, was when they began to escape the shadow of the Irish in Catholic communities and find more economic success as a community, although they still experienced backlash due to their association with Italian fascism throughout the first half of the twentieth century.

As the other reply has indicated, the commonly held idea that the Italians / Irish etc. were not considered white is a result of (sometimes wilful, sometimes accidental) misinterpretation of a sociological theory, which unfortunately has taken hold as the literal truth. I'll explain what we know historically and secondly some academic views on why and how this misinterpretation came about.

On balance it seems that these European ethnic groups were considered white, a fact that is easiest to prove by looking at US immigration and citizenship law, which throughout the 19th century generally required that new citizens be white - and no Irish, Italians, Finns or whatever were excluded under this. However there are many cases of people from Asia failing it, e.g. United States v. Bhagat Singh Thind and Ozawa v. United States. Similarly I'm not aware of the laws against interracial marriage ever applying to someone from Europe.

What is true is that at this time, mere whiteness didn't mean equal status; the WASPS were considered the 'true Americans', to which the European ethnics didn't quite measure up, and this is what accounts for the cartoons comparing the Irish to black people etc. that are often brought up. The only support for the 'European ethnics weren't considered white' view that I'm aware of is a statement from Benjamin Franklin, who in "Observations Concerning the Increase of Mankind, Peopling of Countries, etc." (1755), complained about the increasing influx of German Americans, stating that they had a negative influence on the early United States. The only exception were Germans of Saxon descent "who with the English, make the principal Body of White People on the Face of the Earth. I could wish their Numbers were increased". However on balance I believe that this doesn't describe the general feeling regarding 'whiteness' by the time of the 19th century. Historian Nell Irvin Painter, who wrote 'The History of White People', puts it quite well: "Before the mid-19th century, the existence of more than one white race was commonly accepted, in popular culture and scholarship. Indeed, there were several. Many people in the United States were seen as white — and could vote (if they were adult white men) — but were nonetheless classified as inferior (or superior) white races. Irish-Americans present one example."

So the marriage of your grandparents would not have been seen as a mixed race marriage in the sense that we commonly think of it. However it's definitely possible that the differences in religion, national origin, and wealth could have caused controversy.

A good source on the above is Arnesen, Eric (2001). 'Whiteness and the Historians' Imagination'. International Labor and Working-Class History / Volume 60 / October 2001, pp 3-32.

Now for how this misconception has occurred, drawing partly on the Arnesen above. If you look at the language used in sociology and sociological history, it often uses words in a different way from how they are popularly understood. For example in sociology, 'racism' refers to 'refers to attitudes, beliefs, or behaviors that favor one group over another', without differences in race, as commonly understood, actually having to be involved. 'Minority' (same link) 'is a category of people who lack power, privilege, and prestige in social, political or economic spheres', who 'may actually be a numerical majority'; again this is a different definition from how 'minority' is commonly understood. Similarly in the work of many scholars of whiteness studies, "certain scholars ignore the aspects of identity inherent to whiteness, and "concludes that whiteness is a 'valuable asset' and a 'treasured property' protected by law." (Arnesen 2011), who believe that "Whiteness is not really a color at all, but a set of power relations'" (Arnesen 2001). Basically these scholars are using 'whiteness' as a proxy for 'social power'. This is not a completely unreasonable model for looking at US history, but it ignores the fact that the group at the top of the social hierarchy has not always defined itself simply as white, but has used various markers, such as being of Anglo-Saxon descent, or protestant religion, as well. Unfortunately when these concepts are brought out of the realm of sociology, the academic context is not understood, and people come to the belief that groups like the Italians and Irish were seen as being non-white, as opposed to a dominant group who defined itself as white.