If it were me, I'd take the road in all cases, but here's the long explanation.

Is it an actual separated bike path or is it a bike lane on the side of the road? If it is not separated by a physical barrier from the road absolutely don't ride on it against traffic..... very dangerous and in some states illegal. Even if it is physically separate from the road and riding on it would be allowable, you did say it is close to the road. If you wish to use it, do it with the awareness that it would be like riding against traffic on the sidewalk - one of the more dangerous things you can do on a bike. Do not expect that drivers pulling out of side streets onto the road you are riding will even be looking for you, much less see you and proceed very, very cautiously.