I am using an old version of HC10 and configured PST feature in it. PST files are deleting automatically which irritates me because sometimes I want to keep them for a long time. Is there any way I can specify the time in HC panel for the deletion of PST files or can restrict this automatic deletion?

Yes, this feature is already available in HC10.15 build. You can update the build on your server by following the steps as mentioned in the kb article given below :
https://portal.hostingcontroller.com/kb/a168/how-to-modify-hc-with-latest-build-.aspx
 
Once you have updated HC10 latest build then follow the steps given below for enabling this option in HC10 :

1. Login as Global Admin user in HC10.
2. Navigate to the menu : Configurations >> Manage Servers >> Edit >> Exchange Server.
3. You will find 2 options here namely "Delete PST File After x number of hours" and "Delete After Download".

You can make these settings on your Exchange Server role in order to control the deletion of PST file on your server. If this instructions helps you then you can mark it as an answer as this will help other customers too who are looking for the same option in HC panel :).