Going on Elternzeit does not affect your Blue Card. You still have your job and that is the requirement for a Blue Card.
A problem which may arise is to do with how much Elterngeld you will be receiving. As long as this is enough to cover the living costs for you and, at a later point, your child, there is no issue. If, however, it is below the minimum required amount to live on then it can affect your residency status. Be aware that you will be entitled to Kindergeld. Depending on your income, you may be entitled to additional child supplement (Kinderzuschlag) which does not affect your residency.
As for the Niederlassungserlaubnis, in my view you would not be entitled to it during the Elternzeit. Assuming the income you have is enough to cover you and your child, then the key requirement is to have worked for 21 months. In German, the laws refers to 'eine Beschäftigung ausgeübt'. As I understand it, this means not just to have a job (i.e. a job contract) but to be actively working at it for 21 months. However, I haven't been able to find any concrete information as to whether a person in Elternzeit is still considered to be actively working.
You can apply for the Niederlassungserlaubnis and either the Ausländerbehörde accept you have fulfilled the 21 months and issue it or you will stay on the Blue Card (with the key requirement of being able to support yourself and your child financially).