Can a parent apply for a Non-Immigrant O visa (Guardian) separately from their child's visa renewal?

Yes for the initial Non-O visa and the very first 1-year extension applications. A parent can apply for those anytime after the child is on a Non-ED visa and extension. BUT, the parent's 1-year extension from Non-O dependent on a foreign child with Non-ED/extension is only until the validity of the child's extension.

According to your example, when the child's extension validity is until March, a parent can apply for an initial Non-O visa in May and then apply for a 1-year extension in August, but this extension is until March next year. This is because the "dependent" cannot get a longer stay than the main applicant. After that parent can apply for a 1-year extension in the same time frame as the child.