This code does not directly show the reason for crashing. Please post actual snippets next time.
Also, when posting about a crash, it is good to know how it is crashing. I assume it is a KERN-EXEC 3 panic. But please be explicit about it yourself.
I guess the aHBufPtr passed to StartParsingL is either zero or otherwise does not point to a valid HBuf8 object. That would be the immediate reason for the crash. Why it is not a valid pointer is not visible in the code, but one reason could be shadowing a member variable with a local variable as suspected by @Will.
Some further points, not related to crash here but Symbian C++ in general:
When passing descriptors around, you should use the most general read-only TDesC or TDesC8 type references or read-write TDes or TDes8 type references. Use HBufC or HBufC8 pointers only if you are transferring ownership to the callee.
In this snippet:
bufferPtr.Copy(aHBufPtr->Des());
Copy accepts a TDesC& so you do not need to call Des:
bufferPtr.Copy(*aHBufPtr);
Though if you just want to copy a descriptor, you could use any of the descriptor Alloc functions such as AllocL.