How can my brother and I convince my aging parents to move closer to us?
How can my brother and I convince my aging parents to relocate closer to where we live so that we can care for them better?
You may not be able to persuade your parents to move, if they really don't want to. All you can do is make your best case, and then it's up to them.
A few suggestions: you didn't explain why your parents don't want to move, but it's possible that they are afraid of being a burden on you and your brother. If you suspect this is the case, make it clear that having them closer would actually make life easier for you, since you wouldn't have to worry so much about them.
You could also suggest a trial run: they could try a short-term rental in the area, and see how they like it. This will allow all of you to get a taste of what it would be like to have them living nearby.
You didn't mention your parents' health status, or whether they have plans for their long term care, if and when their health deteriorates. You should discuss this with them, if you haven't already. It's possible that they will be open to moving closer to you when they can no longer live independently. You may be able to persuade them to come for a visit and look at facilities in your area. Some care facilities have waiting lists, so it's a good idea to plan ahead and get them to sign up in advance, if this is something you might consider in the future.
If they flatly refuse to consider a move, you could consult a geriatric care manager in your parents' area, and check out other local resources for seniors. A geriatric care manager can provide invaluable support for people caring for elderly relatives from a distance. Even if your parents don't need it yet, developing a relationship with a geriatric care manager will help ease your mind about how you'll manage when they do.
As you and your brother have grown up, your parents have rationalized the distance/absence with thoughts of "you have a life", "you are busy". And that has worked well for you. Now that you recognize their needs, it is hard for them to insert themselves back into your daily life. Change is hard, for most everyone and discussions of radical shifts will scare them. Moving closer to you doesn't guarantee you will have more time for them. So start small - daily phone calls, Skype, frequent visits will all help you plan your own schedule and will help them become used to your involvement. Sometimes being closer is the first step to living closer.