"Parama means ultimate. Hamsa means swan. So paramahamsa means the perfect hamsa.Hamsa. It is said that if you... Hamsa means swan. If you give to swan milk mixed with water, she will take the milk part and leave aside the water part. Similarly, a person who knows what is this material world... Material world is made of two natures -- the inferior nature and the superior nature. The superior nature means spiritual life, and inferior nature is material life. So a person who gives up the material part of this world and takes only the spiritual part, he is called paramahamsa. Paramahamsa. Spiritual part means one who knows that whatever is working in this material... Just like this body -- your body, my body. Anyone who knows that this movement, the activities of this body is due to the soul which is within this body... That is the real fact. This is only outward covering. Similarly, one who knows that Krishna is the center of these all activities, he's paramahamsa. He knows the fact.
"So bhakti-yoga is for the paramahamsa, one who knows that Krishna is the central fact. Aham adir hi devanam. Mattah sarvam pravartate. So one who knows that Krishna is the cause of all causes, not only theoretically, but practically, convinced, he is paramahamsa. So Kuntidevi says that 'You are meant for the paramahamsas, not for the rascals and fools. You are meant for the paramahamsa.'"
(Srila Prabhupada lecture, New York, April 12, 1973)