Feature Request: Native Serialization for ParameterExpression in QPY

[kiruthikpurpose]

What should we add?

QPY currently relies on Symengine's serialization format for ParameterExpression, which has become version-specific with the recent Symengine 0.13 release. This can lead to errors when loading expressions across different versions. To improve stability and compatibility, we need to implement a Qiskit-native serialization method for ParameterExpression in QPY version 13. This will prevent version mismatch issues and eliminate reliance on Symengine.

Additional Improvements

1. Enhanced Error Messages: Improve documentation and error handling for serialization issues.
2. Backward Compatibility: Develop a compatibility layer for loading older serialized payloads.
3. Performance Optimization: Benchmark and optimize the new serialization method.
4. User Feedback System: Create a mechanism for users to report issues or suggest improvements.

[Cryoris]

This is indeed already tracked in #13252.