Laird shot a second round 68 in San Diego to stay within reach of KJ Choi and Gary Woodland, who were nine under.
England's Rose and Paul Casey missed the cut by a shot, along with American and world number four Rickie Fowler.
American Mickelson was a further shot behind, while Australia's defending champion Day missed out by three.
World number two Day's preparations were hampered by illness and the he missed the cut for the first time in nearly eight months.
Mickelson finished with three straight bogeys as his challenge was ended.
"I think I tried to force the issue a little bit," he said. "When I wasn't under par early, I kind of started to press a little bit. But that stuff happens."

Sexual orientation details are stored by the MoD but kept anonymous, and recruits have the option to "prefer not to say".
A spokesman said the MoD, which introduced the new measure in November, "encourages diversity at all levels".
Openly-gay soldiers have been allowed to serve since 2000.
The spokesman said: "The MoD proudly encourages diversity at all levels. Service personnel are now encouraged to declare their sexual orientation.
"Although this is not mandatory, collecting this data will give us a better understanding of the composition of our armed forces and help ensure our policies and practices fully support our personnel."
Recruits are also invited to say how open they feel they can be regarding their sexual orientation.
The gathered information is not visible on individual personal records or to chain of command or managers, with names removed before the data is collated to make sure no one can be identified.
The MoD said there was no specific drive to recruit more lesbian, gay or bisexual personnel.

Ms Swinson's campaign spending came in £210 below the legal limit, but reports say this was only after almost £7,000 of costs were disregarded.
The SNP said there were "serious questions", with the Lib Dems "living dangerously close to spending limits".
The Lib Dems said they had complied with all Electoral Commission rules.
Ms Swinson re-took the seat from John Nicolson by a margin of 5,339, a 10% majority, having previously lost out to the SNP politician in 2015. She was subsequently made her party's deputy leader.
Her official declaration of expenses was within legal limits, but only after £2,700 of leaflets and other material bought by the party went undelivered - thus not counting towards the total.
A further £4,040 of costs were registered as "national" spending, promoting the party generally rather than Ms Swinson's local campaign.
The SNP said voters would "find it hard to believe that so much cash could be spent on leaflets that just vanished", adding that: "It was an election, not an illusion show."
A spokesman said: "These revelations raise serious questions over the Lib Dems' campaign in East Dunbartonshire. People have a right to know what has gone on here. They have been living dangerously close to spending limits - and seemingly writing off thousands in costs to keep them within the rules."
However, a spokesman for the Scottish Lib Dems said all spending was above board, and said the SNP were "smarting" after losing 21 seats in the elections.
He said: "All Liberal Democrat candidates follow the clear guidance they are given about election spending, and our party's compliance team are in constant dialogue with the Electoral Commission to ensure that we are abreast of the latest advice and abide fully by all their requirements.
"All expenditure in this election was apportioned correctly and clearly identified in our election return which is now a matter of public record.
"It's understandable that the SNP are smarting from a disappointing result, but they should accept that they were beaten in a fair fight in seats right across Scotland, and concentrate instead on the important business of running the country."
Scotland's Martin Laird moved to within two shots of the lead as Jason Day, Phil Mickelson and Justin Rose missed the cut at the Farmers Open.
[NEXT_CONCEPT]
The Ministry of Defence is now asking recruits to the armed forces whether they are gay in a bid to improve diversity in the military.
[NEXT_CONCEPT]
Questions have been raised over the general election expenses of Liberal Democrat deputy leader Jo Swinson in East Dunbartonshire.