How do I analyze a program's core dump file with GDB when it has command-line parameters?

You can use the core with GDB in many ways, but passing parameters which is to be passed to the executable to GDB is not the way to use the core file. This could also be the reason you got that error. You can use the core file in the following ways:

gdb <executable> <core-file> or gdb <executable> -c <core-file> or

gdb <executable>
...
(gdb) core <core-file>

When using the core file you don't have to pass arguments. The crash scenario is shown in GDB (checked with GDB version 7.1 on Ubuntu).

For example:

$ ./crash -p param1 -o param2
Segmentation fault (core dumped)
$ gdb ./crash core
GNU gdb (GDB) 7.1-ubuntu
...
Core was generated by `./crash -p param1 -o param2'. <<<<< See this line shows crash scenario
Program terminated with signal 11, Segmentation fault.
#0  __strlen_ia32 () at ../sysdeps/i386/i686/multiarch/../../i586/strlen.S:99
99    ../sysdeps/i386/i686/multiarch/../../i586/strlen.S: No such file or directory.
    in ../sysdeps/i386/i686/multiarch/../../i586/strlen.S
(gdb)

If you want to pass parameters to the executable to be debugged in GDB, use --args.

For example:

$ gdb --args ./crash -p param1 -o param2
GNU gdb (GDB) 7.1-ubuntu
...
(gdb) r
Starting program: /home/@@@@/crash -p param1 -o param2

Program received signal SIGSEGV, Segmentation fault.
__strlen_ia32 () at ../sysdeps/i386/i686/multiarch/../../i586/strlen.S:99
99    ../sysdeps/i386/i686/multiarch/../../i586/strlen.S: No such file or directory.
    in ../sysdeps/i386/i686/multiarch/../../i586/strlen.S
(gdb)

Man pages will be helpful to see other GDB options.

Simple usage of GDB, to debug coredump files:

gdb <executable_path> <coredump_file_path>

A coredump file for a "process" gets created as a "core.pid" file.

After you get inside the GDB prompt (on execution of the above command), type:

...
(gdb) where

This will get you with the information, of the stack, where you can analayze the cause of the crash/fault. Other command, for the same purposes is:

...
(gdb) bt full

This is the same as above. By convention, it lists the whole stack information (which ultimately leads to the crash location).

Just skip the parameters. GDB doesn't need them:

gdb ./exe core.pid