That operation would be possible in LEKIN only when the software supports network operations and multi-user input/support. I've checked the official website, the description and all the features, but there isn't anywhere mentioned about collaborative work on the database. The software is working individually on the PC where it's launched without any possibility to add other users from different machines.
The answer to your question would be no, you can't use it in parallel machines and have the same database. Each installation uses its own database to store the data.