Let R be a locally excellent domain of prime characteristic and let R+ denote its integral closure in an algebraic closure of its fraction field. It is shown that the tight closure I* of any local parameter ideal I is equal to (IR+ intersection R). It is shown that locally excellent rings in which all parameter ideals are tightly closed must be pseudorational. Applications to algebraic varieties over fields of any characteristic are developed, including a tight closure method for proving that certain varieties over a field of characteristic 0 have rational singularities. Examples are given to demonstrate the effectiveness of this method. It is also shown that the parameter test ideal behaves well under localization: if J is the parameter test ideal for a complete local Cohen-Macaulay ring R, then JU-1R is the parameter test ideal for U-1R, where U is any multiplicative system in R.