I am a mother of a toddler, also I am a graduate student. My advisor suggested the project I’m working on right now over Skype when kiddo was six weeks old, so my project is almost exactly as old as he is. There’s a few topics I could hit here, but I’ll try to talk about: how to parent while a grad student and how to grad student while a parent.
I. How to parent while a grad student:
This part is short! You parent while a grad student the way you would parent if you were not a grad student, that is, you do your best and feel like you’re drowning in laundry and spit-up (a cute version of vomit! But it’s still vomit!) and sleep deprivation. Not to mention the many needs of an adorable helpless creature whose delicious chubby cheeks and arms require both snuggles and food every two or three hours and endless tedious mind-numbing work of diapering and cleaning and rocking and feeding. Also you compare your horrible laundry and spit-up- covered apartment to the other new moms who have beautiful Pinterest houses and coordinating nursery+layette sets with babies who sleep through the night after a month, and feel awful. Also you’re overwhelmed by advice and opinions and anxiety. Actually it’s a lot like being a grad student, but with more cuddles and less sleep.

This is as sleepy as my eyes get before I’m asleep.
II. How to grad student while a parent:
Convince someone to pay you before having a baby! We grad students aren’t covered by FMLA (Family Medical Leave Act of 1993) since most of us have half time teaching appointment positions, if that, and research doesn’t count as a job. FMLA is worthless anyway; it just ensures that your job will be there when you get back after three months, but doesn’t say that you’ll be paid any amount of money during that time. Believe it or not, you have to keep paying rent when you have a baby! Also, you have to pay for the baby! It boggles my mind/scares me that doctors go back to work six weeks after having a baby, because I do dumb things when I’m waking up every three or four hours to feed a screaming thing, and my decisions don’t affect other peoples’ health or bodies.
Once you’ve convinced someone to pay you, consider asking for more money! I recently met a woman whose partner was in the military, and her department paid her more in order to pay for childcare. My kid went to a fancy daycare for months 3-6 of his life (because I could walk to it in five minutes even when it was 0 degrees out and not worry about baby frostbite), which costs $2300 a month if your kid goes 5 days a week. At the time I made $1700 a month. Like I said, that’s the fancy daycare: the church down the street from me now offers daycare for $1000 a month, and infants are the most expensive. So research daycare/having a parent move in/getting a nanny or au pair($$$, but perhaps more cost effective with more kids). Good thing you’re a grad student and good at research, because you’ll get INUNDATED with data.
This is the advice I got about having a kid by a professor who is also a mom, about how to do research after having the kid:
For the first year, just show up to your office and pretend. Pretend to work, and one day you’ll find out that you’re actually working again.
This totally worked for me! I stayed home with the kid until he was 3 months old, then started coming to the office three days a week. After three more months, I went in to the office four days a week, and to a coffeeshop one day a week, which is my current schedule. Those first three months back I’m not sure what I did. I was very sleep deprived. I must have started this research project then, and looking back at my notebooks I see that I wrote a lot of nonsense (at the time I thought it made sense).
The next three months I wrote more nonsense, but at a faster pace and there are bits of sense peeking through. When baby was about a year old, one day I came in to the office and realized that my mind was back! I’d been sleeping reasonably for a few weeks, had stopped breastfeeding, and could look back at my nonlogical arguments and pull at the bits of logic that had been appearing more and more frequently over the past few months to make something sensical. It was so great! And I hadn’t known my mind had left because I was so sleep deprived that I wasn’t really taking stock of my mental status, so it was a pleasant surprise to be back.

He’s not the most helpful for math research.
Now I’m in more of a rhythm- so my advice is “survive the first year” and then “find a schedule.” Maybe you’re better at not sleeping than I am, in which case it’s really “survive the first few months” and then “find a schedule.” In any case, a schedule is important. It’s important if you don’t have kids, but it’s essential if you do. I used to do a few hours of work on the weekends, and now that is possible but not probable because by the time the kid is asleep, I’m mentally and physically exhausted (I think I am more physically weak in general; I know plenty of people who work when the kids are down and plenty who netflix when the kids are down). Also 6-8:30 a.m. and 5:30-7:30 p.m. are both kid chunks of my day, which means I have a business hours schedule for math. This is probably reasonable to you if you are not a grad student, and some non-parent grad students stick to this too (it’s maybe better for your mental health).
I have hopefully said this before and I’ll say it again, the best grad school and life advice I’ve received:
Swim in your own lane.
I am not a pinterest-er, I am not a marathon runner, I am not a world traveler, I am not a social media maven. But I am a mom, and I’m a grad student, and I’m doing okay (I feel like I should say “I’m doing my best” here but I’m not sure that’s true). I’m swimming in my own lane. You should too, and just ignore all the advice I wrote in this post, which was mostly me talking about my personal experiences anyway.
Wonderful advice, which can be usefully extended to parenting as a mathematician, no matter career level.
“looking back at my notebooks I see that I wrote a lot of nonsense (at the time I thought it made sense).” THIS. It happened to me, it happened to friends. It happens. Accept it and go on, it’s part of the process.
A tiny bit of advice: don’t worry if for a few months or years a large part of your family income goes to daycare/nannies. It’s money well spent (in case you have it).
Finally, you know how on airplanes they tell you to secure your own oxygen mask before attending to kids? Do so. Take care of your own health and well-being, do not neglect it because you’re so busy.