Parasurama avataram was to convey to the world, that one must always respect one's parents words and to show that an unjust clan cannot be permitted to rule. 

He was the last son of Jamadakni Rishi and Renuka Devi and was named Raman. Pleased with his severe thapas, Lord Siva gave him a weapon called Parasu. Raman thus came to be called Parasuraman. 

Kaarthaveeriyan, a vain king cunningly stole Jamadakni rishi's holy cow Kamadhenu.  Kamadhenu could grant any wish.  An angry Parasuraman fought with the king, killed him and brought back Kamadhenu & its baby to the ashram.  But his father was upset for he felt a brahmin has to be patient.  Parasurama went on a pilgrimage for a year to please his father & to atone his sins. 

Parasurama's mother had deep respect and devotion for her husband and worshipped him as her Lord. So pure was her devotion that each day she would go to the river bed and pick up a handful of sand and this would form into a pot. She would be able to fill water in this unfired pot and take it to her husband for his daily pooja.  

One such day, as she took up a handful of river sand, she saw the reflection of a handsome gandharva in the water and she looked up in admiration. This was a step down in her sincerity and she lost the ability to create a pot of the river sand. 

Sage Jamadakni realised through his Gnana Drishti of the happenings and ordered his sons to kill their mother. His sons refused to do so, claiming their mother was even greater than God to them. Parasurama immediately chopped off his mother's head to do his duty to his father. Rishi Jamadkni was overjoyed by his obedience and was willing to grant him any boon. Parasuraman asked for the life of his mother and brothers and also that they have no memory of the happenings. Jamadakni rishi granted him his wishes.   

(There is a belief that it is only Renuka Devi who is worshipped as Mariamman. In the temples for MariAmman, the moola vigraham is only the head placed on a pedestal.) 

Kaarthaveeriyan's sons were meanwhile lusting for revenge.  One day, when sage Jamadakni was alone in the ashram with his wife, they chopped off his head.  Parasurama then decided that such bad blood should not be allowed to run in the ruling class and completely destroyed the kshatriya clan.  (Since Parasurama was an avataram, we must assume here that he destroyed the evil kings.)  Moolagan was one of the kings who escaped Parasurama's wrath and Dasharatha, father of Lord Rama was his descendant. 

(The kshatriya clan that followed - Moolagan-Dasarathan-Alabadi-Theerkapadu-Raghu-Ajan-Dasarathan-Lord Rama)