I’ve got two answers to this.
One comes from my own experience.
The other one comes from realizing that my own experience may not match your experience.
From my own experience, the answer is that, unless you teach really young children, you probably don’t want parents walking into your classroom on the first day.
This can be a little tricky: after all, most parents mean well, and you want to keep them on your side. At the same time, having parents walk into the room where you are trying to establish yourself as a new authority figure can really throw off your game. On top of this, the parents who choose to come inside a classroom tend to be a self-selecting group that is more likely to question you, loudly fill you in on their children’s personal problems, or put you in a weird position where you have to find a polite way to ask them to leave. Preparing a polite-but-firm way of keeping parents outside the door can also be a great gift to your students. Kids deserve a clean slate with their peers, and Mommy coming into the room to “kiss her big, brave ninja good-bye on his first day at his new school,” puts a child at a disadvantage.
An alternative answer is that you don’t have to listen to me on this topic. You might want to ask someone who knows your school culture better than I do.
Your school may already have a policy about this. Or there may be an element of your school culture that leads to a very different answer than the one above. A much better person to answer this question would be your mentor teacher or a trusted colleague, which is why you’ll also find this question on this list of questions new teachers should ask their mentors.