What's the difference between attending an Ivy for 4 years vs. attending community college and Ivy for 2 years each?

Some of the assumptions your parents made are seriously wrong.

TRANSFERRING from CC to an IVY/ELITE is a non-stareter:

It's virtually impossible to transfer into a UPenn or other Ivys from a community college. The transfer admit rate is like 1% at Harvard and perhaps 2-3% at other Ivys. I think the only one that is easier is Cornell, but many don't want to be in Ithaca, NY. Maybe 30 years ago, you could transfer from a CC to UPENN but with average admit rates for all Ivys around 5%, they have no need to take anyone from a community college. This is not the case for say State schools like UCLA. If you go to Santa Monica Community College for 2 years and kick ass there, they have a special arrangement with UCLA to take you and apply all your CC credits toward a degree. With Ivys', the curricula is not at the same level. So you might be an A student at a CC but the course rigor might be 50% easier or less. Even if you were one of the lucky ones to transfer into an Ivy, you might only get 1 or 2 semesters of coursework accepted, not 4. Some Ivys like Columbia for example as a Core Curriculum so regardless of when you start, you have to do the entire Core over which is like 32 credits of coursework.

FINANCIAL AID at top schools caps out at $150,000 to $250,000 not $100,000.

Really good schools have excellent financial aid even for families earning $200,000-$250,000. Harvard, Yale, Princeton might have the most financial aid available because their endowments are between $37-52 billion each. So I would not be surprised that even with $125,000 income you should get 2/3rd- 3/4s of your total costs covered by institutional aid. The worse the school the lower the aid. So you'll probably get less aid from Dartmouth, UPenn, Cornell. It might be less expensive to go to a Top Ivy or Elite than a State College which doesn't give out any aid to middle class or upper middle class families.

The Top colleges have more flexible financial aid than you are aware of so I would not let financial aid hold back your college plans until you know how much you will be getting. For a quick calculation use CollegeBoards Net Price Calculators or individual college NPC on their websites.

Yale has a graduated scale of FinAid for families earning up to $250,000. If you use the NPC on Yales website with your families input I'm sure your aid will be $55,000-$60000 of the total $80,000. This will be no different than attending community college and living on campus.

My advice is that you should apply to the colleges that will give you the most aid regardless of whether it is an Ivy or not. For example, Colgate University or Hamilton College are underrated. They have amazing financial aid and Colgate has ZERO loans for families making less than $150,000. https://www.colgate.edu/admission-aid/financial-aid/colgate-commitment

It's easier to transfer say to UPenn from Colgate than from any community college. And who knows you might actually like it there enough to stay and graduate and go on to grad school at UPenn or some other prestigious school.

The last option I'll mention which very few people do because they are all eager to shoot their shot at Harvard or another Ivy, is to make a solid assessment and see if you really have a solid chance of getting into your Top 2 or 3 colleges. If you do not, then I would suggest you do a Post Graduate extra year of HS at a Private Boarding school. All the Top ones have PG programs and they all have excellent financial aid as well. You will not have to repeat any classes but get to choose from say 200-300 courses, with small class sizes. Many of these are college level courses and teach you to be a better thinker and writer. Successful PGs have up to a 10-20 times admit rate advantage applying to a Ivy over a Senior from a public high school. (Phillips Exeter, Phillips Andover, Choate, Deerfield Academy, Lawrenceville, Hotchkiss are the best ones on the East Coast. It still will be hard to get into those because they have admit rates of 10% or less for PG applicants. They only take about 20 on average.).

Good luck.

To add to some good points mentioned in this thread, your best option will be applying to prestigious schools such as UPenn directly. These schools are ideal because they tend to meet 100% of every applicant's financial need, meaning that you will only have to pay a rate of tuition that you can truly afford.

There are also some other schools which meet 100% of financial need which are easier to get accepted to than the Ivy League - I wrote a CollegeVine blog post listing them. Apply to as many of these schools as you can to increase the odds that one of them will give you an exceptional financial aid package. Hope this helps!