You can always force a specific version with a runtime redirect in web.config (or machine.config), provided the Oracle dll's interface across versions has stayed the same:
<runtime>
<assemblyBinding xmlns="urn:schemas-microsoft-com:asm.v1">
<dependentAssembly>
<assemblyIdentity name="Oracle.DataAccess" publicKeyToken="89B483F429C47342"/>
<bindingRedirect oldVersion="4.121.1.0" newVersion="4.112.3.0"/>
</dependentAssembly>
</assemblyBinding>
</runtime>
From my experience this trick has worked when redirecting from 10.1 to any newer version, but once we ran in mysterious errors when trying to redirect backward (see this), therefore I advise you to better find the exact reason why this is happening - look in your *.csproj which Oracle.DataAccess exactly is referenced and/or disassemble your binaries to inspect also the references of referenced libraries. (It is possible that rather than your code some 3rd party lib you use wants this exact version of ODP.NET).