There are a lot of factors this depends on.
For example: For me, I'm trying to build muscle and gain weight. I'm eating in a caloric surplus. I'm only 19 years old; my metabolism is crazy. I workout 6 times a week for around 9-12 hours a week. I don't want to burn any fat or calories than I already am... So I never do cardio.
Depends on how fast and much weight you want to lose, but if your main goal is to lose weight and burn fat while maintaining muscle, and you're doing solid work with the weights on your Push and Pull days, you can absolutely do cardio on your off days. Especially if you want to lose weight faster.
Also, it depends on you personally and how your body responds, but I would never do that much cardio while trying to build muscle because cardio not only burns fat, but it will hinder your maintenance and gaining of muscle mass, too. So if you're leaning more towards building muscle rather than burning fat, I would limit my cardio and just make sure I have a good diet plan for myself and eat cleanly.
It's really all about balance, knowing your goals, and understanding what works for you. Sometimes you have to just give things a try, asses the results, and adjust accordingly.
For your goals, I'd recommend modifying your split. Rather than
Push, Pull, Rest
you might benefit more from something like
Push, Pull, Legs, Rest
Push, Pull, Legs, Push, Pull, Legs, Rest
So instead of doing cardio, just do more weights. After all, there are tons of studies and articles on why building muscle is actually a better way to burn fat and lose weight than cardio.
One reason being having more muscle speeds up your metabolism. Another being when you train with weights consistently, your muscles becomes more insulin-sensitive, and you body will more eagerly transform and store the majority of the carbs you consume as glycogen in your muscles for fuel, rather than store it as fat.
That's not to say don't do any cardio, but instead use it as a supplement to your weight training rather than it being your main activity. After all, cardio does have valuable benefits for overall cardiovascular health, so even an hour a week of direct cardio with 5-6 days of weight training would be solid.