The start of the second round was delayed four hours because of heavy rain with four players still attempting to complete their opening rounds from Thursday.
Seventy-two players didn't complete their rounds on Friday.
Allenby, the 2001 Los Angeles champion, was three shots back of clubhouse leader Chad Campbell, of the United States, who carded a 65 to move to nine-under 133 in quest of the 864,000 dollar first-place prize.
Tiger Woods played four holes at one-under before darkness closed in. Woods is still lurking and is just four strokes back of Campbell with 14 holes to play in the second round.
First-round leader Brian Davis, of England, was in the last group of the day and didn't get to tee off.
Second-round play was scheduled to resume on Saturday morning, but more showers are forecast overnight and into Saturday as a weather system has stalled over Southern California this week.
The worst-case scenario for organisers is to finish the tournament on Monday which is the US President's Day holiday.
England's Luke Donald, Stephen Leaney of Australia and American Rocco Mediate finished their rounds and are four shots behind the clubhouse leader at five-under 137.
Allenby won four years ago on a similar rain-soaked course, coming out on top in a record six-man playoff.
LEADING SECOND ROUND SCORES: (USA unless noted; play halted by darkness with 72 players still to finish their rounds):
133 - Chad Campbell 68-65.
136 - Robert Allenby (AUS) 69-67.
137 - Luke Donald (GBR) 66-71, Stephen Leaney (AUS) 70-67, Rocco Mediate 70-67.
138 - Michael Allen 68-70, Mark Brooks 70-68, Angel Cabrera (ARG) 69-69, Brett Quigley 66-72.
139 - Tim Clark (RSA) 70-69.
140 - Alexander Cejka (GER) 70-70, Jose Maria Olazabal (ESP) 68-72, Mike Weir (CAN) 67-73.
141 - Nick Price (ZIM) 70-71.
142 - Stuart Appleby (AUS) 73-69, Greg Owen (GBR), 75-67, Justin Rose (GBR) 74-68.
143 - Steve Allan (AUS) 70-73, Carlos Franco (PAR) 70-73, Carl Pettersson (SWE) 72-71, John Senden (AUS) 73-70.
144 - Hidemichi Tanaka (JPN) 71-73.
145 - Jesper Parnevik (SWE) 72-73, Rory Sabbatini (RSA) 73-72, Toru Taniguchi (JPN) 70-75.
147 - Stephen Ames (CAN) 72-75, Graeme McDowell (NIR) 74-73.
148 - Paul Casey (GBR) 74-74.