if your area has strips without curbs, those are the best, as a lot of folks park on them and so they are in poor condition. Owners really don't care if you dig in the weeds and bare patches. Also, strangers and neighbors are likely to park on the open strip, so the homeowner already has resigned himself to a certain lack of control over the strip, and damage to the strip, even if he avoids parking on it himself. If there are bumpers along the road edge to discourage parking, then I avoid it. Some homeowners illegally fill their strip with shrubbery and trees...well, it's illegal here, where open shoulders are supposed to be able to accomodate cars pulling over on it, I don't know about your area.
Strips under trees with heavy foliage are excellent...I don't know about up north, but st. augustine grass here in south florida needs a lot of sunlight. These areas can have big bare patches. Again, almost no one has ever insisted I leave from such a strip. If the tree looks 30 years old, and its a hundred year old residence, then tthere was probably thick grass there when silver coins were in use.
Look for unoccupied houses, empty lots, houses with no cars, houses with all the blinds closed and no dogs, just to avoid trouble. Uncurbed strips by churches are great. If the cops come they will either let you continue or tell you someone called and they'll politely suggest you go somewhere else.
It's definitely worthwhile to search out old streets with parking strips that lack curbs even if you have to drive awhile to get there. In south florida we call them swales, but apparently there's no such term in other parts of the country.