The restriction enzyme is an important device which provides cutting operations of DNA strands to construct a DNA-based computational model such as splicing systems [3]. In this paper, we employ a specific type of restriction enzyme which cut on both sides of their recognition sequences [6], and propose a new DNA-based computational model which has several advantages compared with conventional models. The new computational model is shown to achieve universal computability using only natural DNA-based methods such as annealing, cut, ligation and circular strands without any practically hard assumption. Furthermore, while the generative power of the computational model is shown to be universal, the parsing (accepting) computation ability is more appealed. That is, given any string, the model computes whether it accepts the string, and most conventional DNA-based model have not offer this accepting process. We show that the new computational model efficiently computes the parsing process for context-free grammars and finite sequential transducers.