Does the board of directors of an HOA have the right and/or ability to denounce responsibilities like enforcing the convenants?

The extent of the obligation of the Board of Directors to enforce covenants depends on the language in the Deed of Dedication.  Generally the authority is permissive, not mandatory,but varies from subdivision to subdivision.   However, as a lot owner in the subdivision you usually have an equal right to enforce the covenants.  The covenants that "burden" each lot owners use of the land are generally for the "benefit" of all the other lot owners.  While covenants are enforceable, they are generally not favored by the courts, and the enforceability of the covenants can be lost if not strictly enforced.  In other words, it IS a valid defense to an action to enforce the covenants that other people committed the same violation and were not sued.   Your best course of action may be to sue the offenders yourself and then run a strong campaign to elect new directors that will enforce the covenants.  Of course, lawsuits aren't cheap and if your homeowners association doesn't have the funds to conduct the battles, then the membership of the board won't make a difference.