Generally, an FSA will disqualify you from contributing to an HSA. However, this FSA is somewhat special. Your parent's FSA can cover your expenses but not your spouse's expenses. Therefore, your spouse is not considered to have "other medical insurance coverage", and she can contribute to an HSA in her name, using either the single or family limit, depending on what kind of HDHP coverage she has.
Separately, an HSA can always be used to pay for qualified medical expenses for the account owner, a spouse, or the owner's dependents. It doesn't matter how the money got in the account or what kind of insurance you have at the time the medical expense is incurred; once money is in the account it can be used to pay any qualifying expenses.
The rule about having other insurance coverage disqualifies the covered person from making tax-deductible contributions to an HSA but it does not disqualify paying for expenses out of already deposited funds.
