I want to know if our parents will only be our parents in every birth cycle ?
The whole process of taking birth is very difficult and uncertain as the following shlokas show:
Residing in that (region of the moon) till they have exhausted (the results of action)  they then return  again  the same way as they came (by the path that is being mentioned).  They come to Akasa, and from Akasa to air. Having become air, they become smoke. Having become smoke they become the white cloud.
Having become the white cloud, they become the (rain-bearing) cloud. Having become the cloud they fall as rain. Then they are born in this world as rice and barley, herbs and trees, sesamum plants and beans.  But the release from these is more difficult,  for whatever eats the food and sows the seed, they become like only.
Chandogya Upanishad 5.10.5–6
Swami Swahananda, the translator, makes the following points about these 2 shlokas:
 The soul who returns from the lunar world does not do so having exhausted the entire store of this past Karma, but only that part of it which led to the attainment of that region is used up there.
 When the results of action are exhausted the aqueuous body of the region of the moon takes the subtle form like Akasa and remains in the sky. From the sky it successively appears as air, smoke, white cloud and (rain-bearing) cloud and then falls down in the form of showers of rain.
 The word again shows that he has been through all these stages before.
 There is no definite rule that they must come back passing through exactly the same stages as they want. In fact there are differences in the ascending and descending paths. Here one particular path is being mentioned.
 That is, they remain attached to the corn, etc. While mentioning the descent of the displaced souls the text refers to them with a singular verb, but in the last instance a plural verb is used. This is justified as follows. In the first instance they are taken as a unity represented by cloud etc., in which they dwell indistinctly. In the second case, the souls with residual Karma are many.
 There is no certainty where the rain-water will fall and where the Jivas attached to them will go. Moreover, the food along with the Jivas must be eaten by a man capable of producing children, then only can they enter the mother’s womb. To be attached to corn, etc. itself is difficult. To be born as corn etc is for some the last stage of this birth, not an intermediate stage before becoming man etc. In the form of corn etc. they exhaust their Karma and then are born again in another form.
Note of caution: Jivas only get attached to various food products like rice and barley and are not themselves rice and barley or corn.
Thus chances of having the same parents in rebirth are miniscule.
In some special cases, it might so happen for some births only, due to the effect some boons may be.
My reference here is from the Yoga Tattva Upanishad which is linked to the Krishna YajurVeda. It is counted among the 108 Upnishads.
131(b)-134(a). That breast from which one suckled before (in his previous birth) he now presses (in love) and obtains pleasure. He enjoys the same genital organ from which he was born before. She who was once his mother will now be wife and she who is now wife is (or will be) verily mother. He who is now father will be again son and he who is now son will be again father. Thus are the egos of this world wandering in the womb of birth and death like a bucket in the wheel of a well and enjoying the worlds. 134(b)-136(a). There are the three worlds, three Vedas, three Sandhyas (morning, noon and evening), three Svaras (sounds), three Agnis and Gunas, and all these are placed in the three letters (OM). He who understands that which is indestructible and is the meaning of the three (OM) – by him are all these worlds strung. This is the Truth, the supreme seat.
From the highlighted verses we can conclude that the girl who is your wife in the current birth might have been your mother in the previous one or vice versa.
So, same souls getting born as the parents in every birth is not really possible.