This paper presents algebraic identities and algebraic query optimization for a parametric model for temporal databases. The parametric model has several features not present in the classical model. In this model a key is explicitly designated with a relation and an operator is available to change the key. The algebra for the parametric model is three-sorted; it includes relational expressions that evaluate to relations, domain expressions that evaluate to time domains, and boolean expressions that evaluate to TRUE or FALSE. The identities in the parametric model are classified as weak identities and strong identities. Weak identities in this model are largely counterparts of the identities in classical relational databases. Rather than establishing weak identities from scratch, a meta inference mechanism, introduced in the paper, allows weak identities to be induced from their classical counterpart. On the other hand, the strong identities will be established from scratch. An algorithm for algebraic optimization to transform a query to an equivalent query that will execute more efficiently is presented.