You must have both 32-bit and 64-bit Oracle DB Client installed on your machine. What is happening is that 32-bit ODP.NET looks for its dependent DLLs and finds the 64-bit Oracle Client ahead in the Windows Path. What you need to do is set the 32-bit ODP.NET DllPath in your machine.config file to point to where 32-bit Oracle Client bin directory is.
ODP.NET DLL search order:
Sample for setting DllPath in .NET config: