Using fork() in C?

I'm fairly certain that whoever assigned you this homework is looking for the exec() family of functions, from the POSIX API header <unistd.h>, because there is nothing else that more closely resembles the sort of functionality you describe.

The exec() family of functions executes a new process and replaces the currently running process address space with the newly executed process.

From the man page:

The exec() family of functions replaces the current process image with a new process image.

It's not exactly the same as "terminating" the parent process, but in practice it results in a similar situation where the parent process address space is erased (replaced) with the address space of the child process.

What Linux library function is like a fork(), but the parent process is terminated?

The parent process should not terminate because , it must wait for the child processes to finish executing , after which they will be in a state called "zombie state", now it is the responsibility of the parent to clean up the leftovers of the child process. The parent process can terminate without cleaning up the child processes, but then, it is not a proper way to do it, as the exit status of the child processes should be collected and checked by the parent process.

Here is an example, to demonstrate , what i just said...

#include<stdio.h>
#include<unistd.h>
#include<sys/wait.h>

int main()
{
  pid_t cpid = 1 ;
  int status;

  cpid = fork();

  // Load a application to the child using execl() , and finish the job

  printf("Parent waiting for child to terminate\n");

  int wait_stat = waitpid(-1,&status,0);     // Parent will hang here till all child processes finish executing..
  if (wait_stat < 0)
  {
    perror("waitpid error");
    exit(-1);
  } 

  // WIFEXITED and WEXITSTATUS are macros to get exit status information, of the child process

  if (WIFEXITED (status))          
  {
  printf("Child of id %u cleaned up\n",wait_stat);
  printf("Exit status of application = %u\n",WEXITSTATUS(status));
  }

}