The development offices at most boarding schools and top colleges are very active. Even a school like Harvard with a a $30+ billion endowment has active fundraising.
Although the development office and admissions office operates independently there are definite links between the two. Most schools have regular meetings between development and admissions to review potential candidates. Clearly unqualified candidates are tossed out, but those who are within the range, and with parents who can donate, are given priority. The extent of this advantage varies from school to school. Some schools like Caltech offer no advantage, whereas others it can be a huge tip. Another link is that the college counseling office at some of the big-name boarding schools pass along information about potential development cases to colleges.
Now how much of a donation would get their attention in the development office? I honestly don’t know. I worked on the fundraising group for my son’s BS. The average donation for the annual drive from parents was approx $200. They also care about consistent giving. One of the key thresholds is if you can donate enough to self-endow a fund, scholarship or chair. The minimum required would be a few hundred thousand dollars.
Author Daniel Golden wrote a good book on this topic. He gives several examples of kids from well known boarding schools (Pingry, Sidwell, Harvard-Westlake, Lawrenceville) who got into top colleges because of their parents ability to donate. The name of his book is called The Price of Admission. The book is about 10 years old, but gives a good perspective.