The County does have specific code restrictions for new road construction in a stream area. You should consult with the County department of planning or development, and they can tell you the specific code restrictions. There may be issues with the stream as a natural area, and/or as a flood plain. Depending on the rules and the specific purpose of the road it may be prohibited or only allowed under specific conditions such as in the case where it is the only access you can get to the property. Depending on how the stream is categorized the California Department of Ecology and/or the U.S. Army Corps of Engineers may also have permitting roles. Start with the County and they can probably tell you what permits are required by them and other jurisdictions.
It depends on the zoning of the area and any land use restrictions. Be careful however. There can be serious penalties in California for interfering with an established waterway. Check with your city or county.