You should check the SQLite docs about whether that library supports multiple simultaneous connections to the same database file. I just found this very nice summary about applications that should and should not use SQLite
The sections there about client/server applications and high concurrency lead me to think that SQLite and multi-threading are not a good combination. The latter section specifically says SQLite "will only allow one writer at any instant in time."
Update: the bottom line is: either you have to complicate your own script in order to overcome SQLite's limitations (choroba's reply below suggests there's already a CPAN module to help with that), or else you have to pick a more capable type of database.