First I'm going to go out on a limb and guess this is your first parental leave. Because no one in 20 years has asked me that on their second (or beyond) maternity leave unless there was a really long gap. That should tell you all you need to know right there.
The short answer is no. If your ideal vision of your family is a happy supportive partnership, children, and a loyal dog or two, you have a much greater chance of making that happen if you take it one step at a time.
New family members, furry or human, are one of the most wonderful things in the world. And I believe every one is a miracle. But they are no picnic. This is true of new babies, new dogs, blending families, adding foster children, and even marriage. Integrating new lives together, no matter how much love, is A LOT of work. So, to the extent you have the choice, do it one at a time for everyone's sake.
Here's why getting a puppy on parental leave sounds like a great idea but isn't.
1. You won't be sleeping much
Remember, sleep deprivation is a form of torture. You don't know which end is up sometimes. So walking, training, exercising, and keeping a potty schedule for a new puppy are going to be somewhere between difficult and impossible. It's true some babies are better sleepers than others. But I don't know anyone who wasn't sleep deprived on their parental leave. And some babies (like my own) are really bad sleepers. You don't know which one yours will be.
2. Babies eat literally over half the time
In a 24 hour period you are liable to be spending over 12 hours feeding during the newborn stage. Remember, in utero your baby was feeding 24 hours in a 24 hour period. They gradually ease out of this after birth. But in the beginning they feed slowly and over long periods. Exercising, training, and keeping a potty schedule, cleaning up the yard for a needy puppy (dog babies are needy too) are going to be brutal if they are possible at all.
3. Your relationship is in for a wild ride
Over 90% of couples report their relationship experiences strain during the babies first year. And if the non carrying partner is the one pushing this dog idea because the carrying partner will be "on leave" I hate to tell you, you are likely to be one of them. Most relationships do recover and evolve after about a year. But it is hard. Don't make it harder than it has to be by adding a stresser and work load (as adorable as that stress and workload is) that can wait.
4. You, your baby, and your partner deserve all the tender, loving, special moments you can get during this time.
You will remember and cherish the special moments for the rest of your life. And they will carry you through the hard times this season brings. Even if you DO have capacity for a puppy, that is not the best thing to do with the moments you have. Any moment that isn't actively consumed with baby care or sleep should be used for self care, baby snuggles, or partner time. And your puppy deserves to come at a time when they can get your attention and work.
