This is an old, abandoned road.
When the new bridge was built to cross in front of Folsom Lake, the roadways were changed to provide access to Folsom Lake Crossing Road. This strectch of the street was reclaimed by the weeds and shrubs, but you can still make out a hike from about Hancock Street to Briggs Ranch. There is access through the new wall from Takana Drive; take Hancock to the first left, Sombrero, then an immediate left onto Takana, and you will see the gate through the wall. A new brick was was recently constructed there.
It is possible to park off-road from E. Natoma Street, but it may be illegal, as there are No Parking signs along that part of the street.
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