If it had no leaks it would not be loosing coolant. So therefore it does have a leak. It can be leaking in a spot where the coolant burns away before dripping on the ground. Also a blown head gasket is a possibility. Look for coolant in the oil. A heater core can leak. Do you smell coolant inside the car or are the windows fogging up or developing a film. You need to have a pressure test run on the cooling system and find this leak.
Several possibilities - from easiest to most expensive: Faulty or inoperable radiator cooling fan - causing overheating. Faulty thermostat - causing overheating. Water pump failure - causing overheating. Blown head gasket - allowing exhaust gasses to escape into the cooling system and expelling the coolant.
Is "all new coolant parts" everything? Thermostat is the first thing. Is the cooling fan coming on? New radiator cap? If everything is in fact new, take the cap off and start the car to see if the radiator is being pressurized with exhaust. A bad head gasket, cracked head or block could be the problem. Starting it with the cap off will cause coolant to blow out of the radiator.
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