Really should be described as "Obtained all parts for Mobile suits". You need one of each part (rank doesn't matter) to acquire a suit. Some suits don't have legs, so don't freak out about the ball or zeong not having them.
Collection missions help with parts, but some suits (like char's Z'gok and the MP Axis Bawoo) Will need special missions or the luck of getting the right grunts in your mission. You should get a part from every ace you shoot down, and random numbers of enemies for grunts. Parts are shared by all pilots, so no need to stick on one character to obtain them.
You can check your progress in the Mobile suit lab under the first option, selecting a suit, and looking at parts. RB/LB will cycle through and non-highlighted parts will make themselves evident.