1201 Euclid Avenue between Eunice Street and Bayview Place
Codornices Park in Berkeley has a fenced in tot playground, a big grassy field and a reservable table near the tot playground. But if the table is already reserved, many families will set up a table on the grassy field. (the area around the table is near a very shallow stream if there is any water there these days with rocks at the stream side so the little ones are best playing on the grass or tot lot.)
Cordonices park has nice reservable areas, but they are often booked in advance. I had a baby shower there, and we just showed up ahead of time and set up tables in a non-reservable area and had a great time. Its such a beautiful park.
It sounds like you want easy in- town access so my first suggestion is Codornices Park. There are BBQ grills at the end of centerfield (off the grass) of the baseball diamond. More special is to go from there across the creek to the picnic area at the big chimney. Berkeley req does reserve for you, but I'd think not necessary unless a weekend. Another close by, but small and private location is Remoullard Park on Regal Road above Euclid
Our more favorite suggestion would be Tilden and specifically the Spillway Picnic Area. It is east of the great lawn at Lake Anza. Their parking lot was closed last month...but parking in the over-flow lot next to it is actually closer to the BBQ sites. Seven tables, lovely location, short 2 minute wLk from over-flow parking lot (gravel). If you don't know Tilden, easiest directions is from top of Spruce Street to Anza Lake. You also go right by the Merry-Go-Round and could easily stroll between the two. Good luck.
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Cordonices is a lovely park in Berkeley, with plenty of free street parking. You can reserve the tables throught the city of Berkeley. Great playground and really great big slide, and lots of space to run around.
Cordornices Park - Euclid above Cedar in Berkeley Lots of swings, play structures, woods, and a tunnel under Euclid. Has a small enclosed play area just for little ones.