Judging from a quick search online, this isn't all that unusual a requirement from US banks. So far, I haven't seen it here in Canada, and I'll be rather annoyed if TDCanada Trust get any bright ideas.
It's unclear as to whether their bill-paying apps require third party cookies (ie: not hosted on the bank's servers?) or whether this is third party tracking for which the bank has contracted out. Either way, for those of us who prefer not to be tracked by anyone and everyone, this poses a problem. I don't see any way of making Safari/Cookie Manager only accept 3rd party cookies for one site - someone asked exactly this on the Apple forums a while ago and there was no way to do it.
Here's what I'd suggest:
1. Use Firefox for the banking site - you can fill in exceptions in the privacy settings which should do the trick. To figure out which exceptions you'll need to fill in, clear all your cookies, change the settings to accept 3rd party cookies, then go to your bank's online bill pay section and then check to see which cookies have been added. Add those to the exceptions list. I think that should do it.
2. Send an email and/or snail mail to your bank - preferably someone higher up - to complain about this ridiculous policy! Clearly, your bank has changed their online billing app to use a third party cookie (because the actual app isn't hosted on the bank's servers?) or contracted with some outside source for tracking purposes or something similar. Whatever they're doing, it's a change from previous policies and they should, at the very least, be required to provide you with an explanation. I wouldn't expect a miraculous change in policy/practice though - as I noted, this doesn't appear to be unusual for US banks.