Note: Added two photos to the bottom, or integrated parking and trees.
In situations where there isn’t any space for trees along the street, usually where a sidewalk buffer (planting strip) was never provided and where a reconstruction of the street to add sidewalk buffers is not in the budget or possible in the right-of-way, trees can be placed in the parking lane. I am not suggesting here that the entire parking area be replaced with trees, but there some trees and their associated shade for walkers and traffic calming effects could be provided on any street with existing parking.
Portland (PBOT) has a sheet about street tree enhancements, which includes Tree Planting in the Curb Zone:
Tree planting in the curb zone allows for encroaching into the on-street parking zone to increase planting widths. This offers an alternative method for increasing tree well size without negatively impacting people walking.
This new tool provides an opportunity to plant trees along curb tight sidewalks or where the furnishing zone is too narrow for large street trees, locations where tree planting would not be possible under current guidance.
PBOT Pedestrian Design Guide

San Francisco has a Parking Lane Planter page:
SF Better Streets
Parking lane planters are landscaped sidewalk extensions placed between parking spaces at regular intervals or at specific locations. They provide space for street trees and landscaping on streets with narrow sidewalks, where tree planting is limited by conflicts with utilities or driveways, or where there is a desire to visually narrow the roadway.
It does not seem as though most cities have policies about placing trees in the street, and those that do, do not seem of long standing, but certainly the practice exists. Street trees in general, though, are of long standing, with every city having policy and design guidance. Sometimes urban forestry and transportation policies and transportation are well integrated, but as often, not.
The City of Sacramento does have an inventory of trees on city property, which includes planting strips (sidewalk buffers). I don’t know of any trees in the parking lane in Sacramento.
Two readers pointed out locations in Sacramento where parking and street trees are mixed in. Both of these were designed this way; the trees were not added later. Both are on R St, the first with a housing development, with parallel parking, the second with housing and commercial development, with perpendicular parking.

