Are the tenses in this sentence correct:
We have placed order for required parts and waiting to get them.
You made a good start but there are some problems.
We have placed (1)an order for (2)the required parts and (3)are waiting to get them.
I have put my corrections in bold and italic type.
(1) You needed to put "an" (one of the indefinite determiners)before the word "order". See this link for why you needed to do that.
(2) You neglected to use the definite article "the" before "the required parts." When you are talking about specific nouns, you need to use the definite article "the". Here is a page where you can read the specific rules on the definite article.
(3) You left out the word "are" which is part of the verb form you are using. This verb form is called the present participle.
Added after the question was edited
The question was edited while I was working on my original answer, so I will add to my answer.
Are the tenses in this sentence correct?
You have the present perfect tense in the first clause: "We have placed". It is used correctly. The action (placing the order) began in the past and the exact timing was not specified.
You have "waiting to get them" in the other clause. It is fine to use this tense (the present participle) after the present perfect tense as you have, but you needed "are" to complete the verb form as I wrote in correction number three.
You would fix it by turning it into: We have placed an order for required parts, and are waiting to get them. Those two extra words, and the comma are the only thing you would need to add.