The parameterized matching problem is a variant of string matching, which is to search for all parameterized occurrences of a pattern $P$ in a text $T$. In considering matching algorithms, the combinatorial natures of strings, especially periodicity, play an important role. In this paper, we analyze the properties of periods of parameterized strings and propose a generalization of Galil and Seiferas's exact matching algorithm (1980) into parameterized matching, which runs in $O(π|T|+|P|)$ time and $O(\log{|P|}+|{\rmΠ}|)$ space in addition to the input space, where ${\rmΠ}$ is the parameter alphabet and $π$ is the number of parameter characters appearing in $P$ plus one.