Forget part-time for now. There are lots of two income full-time parents that get by and if that's what you want, then you can do it too. We get by because we have a fabulous nanny. She does all of our laundry and she runs errands while the kids are in school. Before we had our nanny, we were in your situation and this is what we did:
(1) meal prep-- I went to a dream dinners (or equivalent, I think there are a bunch of them) on a Saturday and made a month's worth of meals that I could freeze. The food is good and fresh, and that way I wasn't eating take out every night. Buy a slow cooker. There great slow cooker recipes that require no more than 10 minutes prep, and then your clean up is minimal.
(2) Order everything online (as other posters have said). Plus, the online savings of diapers at Amazon is great -- even better, sign up for subscribe and save program and they are delivered to your door without you needing to order - automatically come once a month.
(3) Laundry - my work wardrobe is mostly dry cleaning. There is a dry cleaning service in our neighborhood. I still like my old dry cleaner the best, so we go there, but my neighbor's love the service. In terms of actual laundry, all of our towels/sheets are the same color. Boring, I know, but makes my life easier.
(4) Rest
You can do this if you want to do so. No, it's not fun, and when our nanny is on vacation, and I'm juggling everything, it can be tiring, but we make it work.
In terms of the house, I have rooms that I declare "off limits" for toys. Your child is young, but if you enforce this rule on yourself and him as you get older, it will make your life easier. I don't limit where my kids play, but I don't want to see a toy in the living room or the dining room, and if they want to bring a toy into play, then they need to take it out when they are done. As for our play room, I could care less what it looks like (well, within reason). All coats must be hung (or, heck, thrown) into a closet when people walk into the door. This way, when I walk into my house, I don't see stuff on the stairs (which drives me crazy) or a mess. There might be mess behind the living room or in the kitchen, but at least I feel like the house is presentable.
What do I give up? Exercise. I used to run every single day, and now I'm lucky if I go out three times a week. I don't read as much as I used to, and I don't sleep nearly enough. I can't remember the last time that I went shopping for clothing for myself.
I am the primary wage earner in my house, and we cannot afford to live without my salary on a full-time basis, so we make it work.