The larger an animal, the harder it is to cool down. Ignoring the problem of bone/muscle strength vs size/weight and saying an animal could get to outlandish sizes, would a second cardiovascular system specifically for thermoregulation help it regulate its core body temperature? What would the fluid in question be, assuming this hypothetical organism is of terran biochemistry?
Reality check both FAILS and PASSES the test...
Life As We Know It: Sticking with life as we know it on Earth, the likelihood for a parallel cardiovascular system to evolve is highly unlikely. Hasn't happened yet, but that won't stop us from considering other possibilities.
But in other worlds, other evolutionary pathways might be chosen. Now it happens to be the case that we Earthlings do have two circulatory systems already. One circulates blood. The other circulates lymph. It's not a closed system, though, but it hints that other parallel systems could evolve elsewhere.
Deluxe Closed System: Such a system could indeed help with thermoregulation. Instead of draining lymph to the core, this system would drain heat from the core to the exterior. An organ, possibly some sort of "lung" that would have lots of surface area, would exchange heat from the CV to the HE system. This latter would simply circulate high temperature fluid (perhaps water or glycol) out to some kind of radiator appendages. These could be fins or frills or wings or fronds. The thin skin of such appendages will facilitate transfer of heat to the environment.
This system could also be used in reverse. If the animal becomes too cool in its core, it could at first reduce its secondary heart rate and then escalate by reducing the surface area of its fronds or frills, perhaps by curling them up or folding them within a pouch of some kind. It could also seek out relatively warm surroundings and, extending its sails and frills into the warmth, would simply pump warm fluid back down to the core.
The heat evacuation system described above is entirely closed. This prevents predators, disease vectors, rubbish and crud of all sorts from gaining access to the animal's insides. It also allows for a more nuanced and quicker response to body vs environmental conditions.
Not so Deluxe Open System: As AlexP reminds us, life as we know it provides other sources of inspiration. Some animals don't mind if a little crud gets in, so we also find inspiration in the water vascular system of the echinoderms. In this system, water is drawn in through an orifice and pumped through vessels throughout the body as a means of locomotion. The WVS is multipurpose in sea creatures, much more like our circulatory system. This kind of system, parallel to the CV system, could evolve in a large warm blooded creature for thermoregulation.
Water from the surrounding sea would be captured via an orifice and pumped through the thermoregulatory system's vessels perhaps via peristaltic action. This cooler water would pass through to the warmer core where it becomes warmed. Once the water is warm enough, it's expelled again through the same orifice and the process can be repeated as needed.
The heat evacuation system described above is an open system. Although safeguard are in place (baleen or hair like filters), some debris and a whole host of microorganisms can still gain entry. Small stones or debris can clog the vessels while microbes are brought into close proximity of vital organs. A more serious issue is that the animal would almost certainly be precluded from inhabiting warm waters: if it can only take in warm water, it will eventually suffer from hyperthermia, being unable to sufficiently regulate its core temperature.
Biological heat pump either way.
I don't see why a separate coolant system would be needed or would evolve.
Animals already have blood pumping around their system and already use it for cooling through panting (e.g. dogs) air cooling (e.g. elephant ears) or evaporation (e.g. humans sweating).
All that would be needed would be to run the vascular system through the cooling (through whatever mechanism) systems and then through the body. Potentially for very large creatures you would have a special arrangement where blood goes to the surface to be cooled, then goes through the body, then back to the surface, etc.
You could also have multiple hearts, multiple heat sinks, etc, so that for example each section of the creature uses one blood system with then links between those systems to equalize pressure. But it would all just be evolved from the very same systems life uses right now.
Let us assume this is a land animal. Heat must be offloaded to the air. Methods for doing this include increasing surface area, and harnessing the respiratory system to move dead space back and forth (panting).
A truly massive creature might aerate its interior by producing a system by which air is routed through the body and out. This might be a second respiratory system which did not concern itself with gas exchange or maybe an adaptation of the digestive system, putting to use existing mechansims for taking in air.
The result would be a wind through the body of the creature. A wet creature like us would probably also saturate this air with evaporated water, losing temperature to the phase change as we do when we sweat.
Another possible coolant is acetone. Acetone is produced by biological processes including our own bodies - the ketogenic diet is named after the ketone acetone that our bodies make on this diet. Acetone has a low vapor pressure. A finger wetted with acetone will feel cold as the acetone evaporates, the phase change carrying away heat. Acetone sweat would be a formidable coolant. A problem is that the acetone costs carbon to make and then is lost, so using acetone to cool would be more metabolically expensive than using water.
Also a creature that sweat acetone would be flammable, but would probably go out quickly.