Should I have a parent table with only one field?
Do not store dates in a separate table unless there is something about a particular date that has additional information that you need to store.  In general the date is a simple attribute of some other entity, it isn't an entity in itself unless you have a very peculiar application.  

One exception to this might be a star schema data warehouse where there is a time dimension table that might use date as the primary key and have attributes that describe the way that date might roll up - assuming that some roll ups might not be trivial to calculate (like fiscal reporting period or some such).

You can index a date column just as easily as an integer column, so there is nothing to be gained by abstracting out a date so that you can give it an integer alias value.

You can also put a date column in a WHERE clause so searching for records by date is not at all onerous to do.