There are many more parks and parking than listed on that Google map.
And, when it gets to AWMA road at Aliso Woods Park, weekends only - you can continue for a few more miles on AWMA road to the sewage plant. (there you are only a mile from the ocean, but you can't go - some do hop the fence)
If you want to go to Monarch Beach, you can go across the street (Alicia Pkwy) at Aliso Woods trailhead to Laguna Nigel Regional Park, ride through the park with minimal traffic to Crown Valley and Niguel Rd.
Climb Niguel Rd (very tough climb on the road) heading SE and catch the bikeway at SE corner of Marina Hills Dr and Niguel Rd.
Follow the bikeway through Salt Corridor Regional Park and end at Salt Creek Beach Park.