The official word from Disney is that you must move your car. Non-resort guest parking is limited to the time you are actually using the resort..eating, shopping, etc. Once you leave the resort, and head to a park, you are supposed to move the car.
Now...having said that....there are times of year that Disney is really hard core about this. Other times..not so much. If the parks are really busy, or there is a special event going on, then they will tell you to move your car.
If you are staying offsite, and you will have to pay to park anyway, just valet park. It will cost about the same as theme park parking. You are then okay to leave your car, all day, and head wherever you wish to. But, there may be times that valet parking is open only to resort guests...but that doesn't happen very often.
If you aren't willing to pay for valet parking, and don't want to worry about your car sitting at the CR all day, then just park at the TTC when you first arrive. Then, take the monorail to the CR...takes about 20 mins once you board the monorail. And, the monorail starts at about 7am.