In most database systems, the values of many important run-time parameters of the system, the data, or the query are unknown at query optimization time. Parametric query optimization attempts to identify at compile time several execution plans, each one of which is optimal for a subset of all possible values of the run-time parameters. The goal is that at run time, when the actual parameter values are known, the appropriate plan should be identifiable with essentially no overhead. We present a general formulation of this problem and study it primarily for the buffer size parameter. We adopt randomized algorithms as the main approach to this style of optimization and enhance them with a sideways information passing feature that increases their effectiveness in the new task. Experimental results of these enhanced algorithms show that they optimize queries for large numbers of buffer sizes in the same time needed by their conventional versions for a single buffer size, without much sacrifice in the output quality and with essentially zero run-time overhead.