We present a computational model for Parsing Expression Grammars (PEGs). The predecessor of PEGs top-down parsing languages (TDPLs) were discovered by A. Birman and J. Ullman in the 1960-s, B. Ford showed in 2004 that both formalisms recognize the same class named Parsing Expression Languages (PELs). A. Birman and J. Ullman established such important properties like TDPLs generate any DCFL and some non-context-free languages like $a^nb^nc^n$, a linear-time parsing algorithm was constructed as well. But since this parsing algorithm was impractical in the 60-s TDPLs were abandoned and then upgraded by B. Ford to PEGs, so the parsing algorithm was improved (from the practical point of view) as well. Now PEGs are actively used in compilers (eg., Python replaced LL(1)-parser with a PEG one) so as for text processing as well. In this paper, we present a computational model for PEG, obtain structural properties of PELs, namely proof that PELs contain Boolean closure of regular closure of DCFLs and PELs are closed over left concatenation with regular closure of DCFLs. We present an extension of the PELs class based on the extension of our computational model. Our model is an upgrade of deterministic pushdown automata (DPDA) such that during the pop of a symbol it is allowed to return the head to the position of the push of the symbol. We provide a linear-time simulation algorithm for the 2-way version of this model, which is similar to the famous S. Cook linear-time simulation algorithm of 2-way DPDA.