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How is it possible for an offspring to exhibit a recessive trait if neither parent exhibited that recessive trait? What must be true about the parents' genes for this to occur?

Even though neither parents express the recessive trait, the parents could be carriers of the trait by being heterozygous for that trait. For example, having an Aa genotype. If both parents are heterozygous and possess one recessive allele, there is a 25% chance that their offspring will inherit a recessive allele from each parent and have an aa genotype. Hence in order for this situation to occur, both parents need to be heterozygous for the trait, ie have an Aa genotype.