Typically, VA Associations are incorporated as a non-profit. Expecting your Association to be typical, in addition to the applicable HOA/COA law, the
VA Nonstock Corporation Act would also apply. Corporate laws typically address administrative issues.
Per that law there is nothing which would prevent two members from the same household from serving. Therefore, you would need to see if there are any limitations in your governing documents that would prevent this from occurring.
If there are no limitations preventing both from serving at the same time, it is likely legal. However, you should ask a local attorney versed in corporate law to be sure.
Being legal is not the same thing as being a good idea.
As you pointed out, when an issue is before the membership for a vote, it is typically 1 vote per lot. However, when an issue is before the Board, it is one vote per Director. Having two individuals from the same household serving on the Board can create a perception of unfairness. Therefore, Associations should avoid this whenever possible. Unfortunately, if there are not enough volunteers, it is sometimes not avoidable.
Hope this helps,
Tim