I'm not downvoting your answer, but the fact that the OP speaks in terms of "accuser" and "broader legal argument" implies a scenario of the HOA board actually filing a claim in court. Accordingly, what you indicate is "the only way the homeowner would be in the position necessary to decide to prove it or not" matches what the OP was referring to. Hence why the emphatic denial with which you begin your answer is misleading or at odds with the major portion the answer.

@Iñaki Viggers you are being generous. The most straight-forward reading of the OP is to view it as a question about whether HOA board has the authority to make up procedures to enforce its rules after the fact that HOA becomes aware of a reason to suspect that the rules have been broken.

@grovkin The reading you propose is hardly straight-forward. The OP's mention of lack of a deadline in HOA bylaws seems clarificatory, yet far from central to his explicit question on who has the burden of proof. Also the author who first posted an answer mentions preponderance of evidence, attorney fees, and other concepts inherent to court proceedings. Even if one adopts your interpretation, this would reduce to the futile issue of whether the HOA board is allowed to instate some sort preamble or dispute resolution method by which the owner is given an opportunity to obviate a suit.

@Kate Point taken, although such uncooperative approach by the homeowner can backfire if the HOA board files suit upon the inquiry. That is because the OP's inexplicable disregard of the opportunity [offered by the board] to obviate court proceedings may be found vexatious, thereby warranting an award of attorney fees.

@Iñaki Viggers futile indeed. The question's spirit is that what's happening doesn't sit well with the person asking it. And it seems to me that this is an attempt to assert an authority which HOA does not, in fact, have. The OP just can't quite phrase correctly why it doesn't sit well with him, but the circumstances, which he describes, seem to point in that direction.

@IñakiViggers while a judge might view this as vexatious, they may also side with the homeowner's view that requests to prove "one's innocence" (i.e., demands to demonstrate compliance) amount to harassment unless the homeowner has previously explicitly agreed to undergo such inspections.

@grovkin The prolonged stay of the vehicle and the lack of homeowner's response to the board's inquiry weaken the homeowner's allegation that enforcement of a HOA rule amounts to harassment.

I am not a lawyer, I have served on the board of my condominium, and we retained a lawyer to advise us on the law as the need arose. @grovkin you seem be expecting that the bylaws of the condominium establish a sort of a manual of civil or criminal procedure. They don't. The section in the bylaws regarding the assessment of fines was all of two or three paragraphs long. In my experience the way these issues work is that someone complains to the board about a violation of the rules,

The board asks the supposed offending party about the problem, crickets, the board asks again, more crickets, the board sends a notice that a fine is being imposed, the party being fined requests a hearing before the board. The board holds the hearing, explanations and or evidence are provided, and the matter is resolved, and the fine removed.

Note though that the board could impose the fine without hearing any response from the "accused" party, based purely on the board's assessment of the situation.