“It was brutal,” Day said, adding: “If he was alive still, he’d be the one thing that would be holding me back. He was one of those guys that would be out here on the range every day. He’d have a lot of input into what I was doing. Whether it was good advice or bad advice, it didn’t matter; I’d probably have to listen because he was very strict that way.”  Day, 28, insists that he will not raise his 3-year-old son, Dash, the same way. In their relationship, he joked, the son is the one setting the target for the father.  “He always says to me before each tournament, ‘On Sunday, make sure you win because I want to kiss the trophy,’ ” Day, a three-time winner in 2016, said, laughing.  His dynamic with his son is different from what he had with his father. “We discipline Dash when we need to,” Day said. “I understand that. But I want to be one of those dads who’s fun to hang out with. I’m more of a kid myself when I’m hanging around Dash.”  The Days’ me-and-my-best-friend vibe comes across loud and dear in a video commercial made by Day’s equipment sponsor, TaylorMade. The promotional spot, coordinated to air around Father’s Day and casting Dash in a starring role, shows video clips of him hitting golf balls and running onto the green to get swept into his father’s arms.  “When I first saw the commercial, especially at the end when he says, ‘I love you, Daddy,’ I just started crying,” Day said, adding, “I replay it over and over and I cry every time.”  Some kids have baby albums; Day’s son will have a video scrapbook of his childhood.  “It’s so neat,” Day said. “He’s going to have those memories to look back on. We’re going to have that commercial forever, and when he’s my age, he’ll be able to see when he was little he ran out on the greens when I won tournaments.”  A rich life is made of such memories. “That’s what it’s all about,” Day said.  For Phil Mickelson, the United States Open and fatherhood are inexorably entwined. In 1999, less than 24 hours after he recorded the first of his record six runner-up finishes in the tournament, his wife, Amy, delivered the couple’s first child, their daughter Amanda Brynn.  From a swaddling infant, his firstborn has grown into an independent young woman. Mickelson spoke with pride recently of a school-sponsored trip she recently took to a remote mountain village in India to teach impoverished children. “I learn from her,” Mickelson said. “I could go on and on about her.”  Three years ago, Mickelson left Merion, site of the 2013 United States Open, to fly across the country to attend Amanda’s eighth-grade graduation in San Diego, returning a few hours before his first-round tee time.  On Tuesday, his middle child, Sophia Isabel, graduated from eighth grade, and Mickelson was there. He flew home on Monday after a practice session at Oakmont and returned after the Tuesday night ceremony. He had a private jet at his disposal, which no doubt made the commute more manageable. But one got the feeling from listening to Mickelson that he would have hitchhiked if that were what it took to be there for his daughter (his third child, Evan Samuel, is two years younger than Sophia).  “I think any parent would, if they had a chance to be at their child’s eighth-grade graduation,” Mickelson said. “I think that’s a big milestone.” He added, “It’s just important for me to be there for that stuff.”  Mickelson, who turned 46 on Thursday, recognizes that traveling three time zones and back in 24 hours is not the ideal preparation for someone trying to complete a career Grand Slam with a victory here. But such is the juggling act that is fatherhood.  “Those are the differences that I’ll have, where a lot of the young guys in their 20s don’t really have to think about it yet,” said Mickelson, who missed the cut after the second round. “But it’s also brought me some of the greatest joy in my life.”  Wilkes-Krier tried between eight and 10 times, he said, to qualify for the United States Open. He skipped the local qualifier last year because it coincided with the birth of his first child. In his first attempt to qualify as a father, he finally succeeded. A coincidence? Wilkes-Krier thinks not.  His wife, Whitney, reminded him of a vow he made last year to their daughter, Azalea. “I picked her up and said, Next year on this day we’ll be at the U.S. Open,” Wilkes-Krier said. With his eyes glistening, he said, “It’s special.”  A version of this article appears in print on June 19, 2016, on page SP8 of the New York edition with the headline: For Many Golfers, Fathers Serve as Caddies of Life. Order Reprints| Today's Paper|Subscribe
The United States Open, which concludes on Father’s Day, is a convenient time for players to recall how they are shaped by their fathers and their own children.