The primary - perhaps, only - real item at issue is, whether or not your employer's plan is also a HDHP plan. It's also good to verify that they're offering to contribute to an HSA and not an HRA; the latter is compatible with any health plan (including non-HDHPs).
If you're covered under your employer's plan, and it is not a HDHP, then you clearly do not qualify for an HSA. I would be a bit surprised if it were an HDHP, given how you describe it, but I'd also be surprised if your employer offered HSA contributions alongside a non-HDHP.
From the IRS document on HSAs (for 2014 tax year, most current available) (Page 3):
Qualifying for an HSA
- To be an eligible individual and qualify for an HSA, you must meet the following requirements. You must be covered under a high deductible health plan (HDHP), described later, on the first day of the month.
- You have no other health coverage except what is permitted under Other health coverage, later.
- You are not enrolled in Medicare.
- You cannot be claimed as a dependent on someone else's 2014 tax return
If you're covered under an HDHP by your employer, and your parents' HDHP, you probably are okay; this was covered in this question , and still seems okay form the discussion of Other health coverage the above quote refers to (page 4):
Other health coverage. You (and your spouse, if you have family coverage) generally cannot have any other health coverage that is not an HDHP.
As far as how the maximums go, you should be indeed covered by the family maximum. Page 6 of the above document:
Rules for married people. If either spouse has family HDHP coverage, both spouses are treated as having family HDHP coverage. If each spouse has family coverage under a separate plan, the contribution limit for 2014 is $6,550. You must reduce the limit on contributions, before taking into account any additional contributions, by the amount contributed to both spouses' Archer MSAs. After that reduction, the contribution limit is split equally between the spouses unless you agree on a different division.
Of course, that's $6550 for the two of you combined; you get all of it if she doesn't have an HSA, but otherwise it's limited by what she elects.
You should also verify that your employer has an HSA and not an HRA. HRA contributions are a way for your employer to do the same thing as the HSA, except they're more limited purpose - typically they pay the first X dollars of your deductible, and don't carry over from year to year.
Finally - I'd also verify that you are still eligible to be covered by your parents' health plans. You may well be, but stay on the safe side there as well; nothing worse than finding out you owe significant expenses because you didn't dot your i's. Also, this is all assuming you're not eligible to be claimed as a dependent by your parents (it sounds like you're not, but, another i dotted).
And of course - this is not tax advice, or intended to be such, and I am not a tax advisor, nor am I your tax advisor.