This is an important question for understanding the system of American higher education. (I do not read it as being an off-topic request for "How do I improve chances of admission to college?" especially since changing a family's wealth is a much more difficult course of action than, say, paying for tutoring.)
There has been some quantitative study of "legacy admissions," with Hurwitz (2011) estimating that the odds of admission are multiplied by 3 for students with a parent who attended the school, after controlling for other factors. (For instance, legacy students in his data tend to have higher SAT scores than non-legacy students.)
He explains that favoring students whose parents attended has benefits similar to favoring those who apply early or sports stars:
Highly selective colleges face the challenge of maximizing the academic profile of their student bodies, with the understanding that sacrificing some academic talent now will enable the college to preserve or improve its selectivity in the future. In other words, from the college's perspective, an exclusive focus on academics in the admissions process is not sustainable. ... Other tradeoffs might include relaxing admissions standards for early decision applicants to decrease acceptance rates and increase yield rates, and consequently to appear more selective (Jensen & Wu, 2010). Or they might involve admitting academically lackluster star athletes to maintain the successful sports teams that encourage alumni giving (Holmes, 2009; Meer & Rosen, 2009). Relatives of alumni (legacies) offer enthusiasm and familiarity to colleges, and the special treatment awarded to them in the admissions process helps to preserve generational ties that also are intended to motivate financial generosity. (from section 2.1)
His data do not include parents' giving rates, but they compare the same student's admissions across multiple colleges, showing an advantage at a parent's alma mater.
For some historical background, in a legal review of legacy admissions, Lamb (1992) explains how legacy admissions policies came about. Harvard and Yale kept out most public school students with a Latin requirement through the end of the 19th century, and only had to deal with selective admissions upon dropping that requirement. Their (at least somewhat) meritocratic process that followed let in "too many" Jewish and Catholic applicants. Columbia started asking for family information (religious affiliation, father's name and place of birth) in 1919, and other elite schools followed. In 1926, Harvard announced it was taking things like "personality" into account in admissions and started requiring photographs with applications; they used the photos and the discretion to make it harder for Jews, Catholics, and (likely) black students to gain admission, while increasing the proportion of legacy students. (p. 494)
So, the example of John F. Kennedy's 1935 Harvard application is an interesting one. He had famous parentage and family background at Harvard (and by WWII 1/4 of Harvard students were legacies, according to Lamb p. 495). In his case, his Catholicism was probably not held against him.
Lamb also reviews data from a Department of Education Office of Civil Rights inquiry into Harvard admissions. Circa 1992, the Harvard admissions process benefited legacies, and gave additional weight to children of alumni who served on the "Schools and Scholarship Committee" (p. 502), which is apparently a group of alumni interviewers. (My guess would be those who give time are more likely to also give money, but I do not know that donations would change the odds of being accepted as an alumni interviewer.)
In the graduate realm, my guess would be that graduate programs that charge admission (especially for professional training) are much more likely to have legacy preferences than academic programs. However, in subjective processes where a department is hoping to get a good yield of applicants or hire a faculty member who will choose them, I'm sure that there is some benefit in being able to say in an interview, "It's great to be back at this university! My parents met here..."
As a backdrop to this question, there is a longstanding literature that admission into selective colleges, attendance at any college, and graduation rates from college, depend strongly on family socioeconomic status, including level of parent education and family income.
For instance, Duncan, Kalil, and Ziol-Guest (2017) find:
Across 31 cohorts, we find that increases in the income gap between high- and low-income children account for approximately three-quarters of the increasing gap in completed schooling, one-half of the gap in college attendance, and one-fifth of the gap in college graduation.
Controlling for family structure does not explain much of the difference they find, but some of it is related to maternal age; the "maternal age gap" between higher-income and lower-income families increased over the time period they're looking at. (It is not clear whether maternal age is causal or a marker of other family differences.)
Chetty, Friedman, Saez, Turner, and Yagan (2017) look at intergenerational mobility by parent wealth, finding that children from rich families are much more likely to attend college--and fancier colleges--than their poorer counterparts.
Family connections and legacy admissions will not shape these overall enrollment statistics, because the kind of students whose families have that pull are very unlikely to be on the margin of not attending any college.
Instead, sociologist Annette Lareau's idea of "concerted cultivation" is probably much more relevant. Middle- and upper-class parents know the pathways for success and train their children into the attitudes and behaviors that prepare them for success in this society. The example aeismail gives of volunteering to work in a lab runs along these lines.