Are you running SSIS packages in 64-bit mode?
Additionally, RAM could be fragmented.
There is a useful RAMMap utility by Microsoft to analyze what is going on with memory.
Download link: rammap
- You can install it on both servers and compare what is going on with memory on both servers.
- RAMMap allows to empty memory via Empty menu entries.
UPDATE
(1) You can try to use SSIS Balanced Data Distributor: ssis-balanced-data-distributor-overview
The OLE DB Destination
(2) What OLEDB Provider are you using?
It is better to use a newer Microsoft OLE DB Driver for SQL Server.
It is available since 2018. Everything else is deprecated.
https://learn.microsoft.com/en-us/sql/connect/oledb/oledb-driver-for-sql-server?view=sql-server-2017
"...
Important
The previous Microsoft OLE DB Provider for SQL Server (SQLOLEDB) and SQL Server Native Client OLE DB provider (SQLNCLI) remains deprecated and it is not recommended to use either for new development work..."