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We consider the problem of supporting payment transactions in an asynchronous system in which up to $f$ validators are subject to Byzantine failures under the control of an adaptive adversary. It was shown that, in the case of a single owner, this problem can be solved without consensus by using byzantine quorum systems (requiring a quorum of $2f+1$ validations per transaction). Nonetheless, the process of validating transactions remains sequential. For example, if one has a balance of ten coins and intends to make separate payments of two coins each to two distinct recipients, both transactions must undergo processing by a common correct validator. On the other hand, these two transactions are non-conflicting as they do not lead to double spending, allowing in principle for parallel validation. In this paper, we show that it is possible to validate payment transactions in parallel with less than $f$ validations per transaction in an asynchronous system, provided that each transaction spends only a small fraction of a balance. Our solution relies on a novel class of probabilistic quorum systems that we introduce in this paper, termed \textit{$(k_1,k_2)$-quorum systems}. In the absence of an adaptive adversary, \textit{$(k_1,k_2)$-quorum systems} can be used to enable concurrent and asynchronous validation of up to $k_1$ transactions while preventing validation of more than $k_2$ transactions. Employing a $(k_1, k_2)$-quorum system, we introduce protocols enabling a payer to validate multiple \textit{fractional spending} transactions in parallel with less than $f+1$ validations per transaction. Subsequently, the payer reclaims any remaining funds through a fully validated transaction, referred to as a \textit{settlement} transaction.