There are parking bays around the area. Best to get there with some time to spare and drive around the area. If for some reason you can't find a place you could park in Hyde Park which is about a ten minute walk away from the Embassy.
Registering births is done in the basement, which involves going down a flight of very steep stairs, so take care with the baby buggy. I assume you will be applying for a passport as well? If so make sure you have wet wipes with you as they take a finger print from the baby. The whole process will probably take about one hour.
Have a look at this thread