Phonetic and semantic similarities lead me to think they might be related. I can't remember where, or if it's even a non-folk etymology, but I read somewhere that pater and partitio are related, in the sense that when one opens or "partitions" the womb, one becomes a father.
Reduction of the problem
It is good to preprocess the data and reduce the words into simpler constituents within Latin. We can reason as follows:
- parens < parere
- parturitio < parturire < parere
- partitio < partire < pars
This reveals that parens and parturitio are already well linked within Latin, both coming from parere. Now it remains to compare pater, parere, pars or their stems patr-, par-, part-.
If you want to assess whether two words look similar, do it at this level. If there are many layers of derivation and morphology, the similarity could be due to that. Strip away any prefixes and derivations you can.
Perhaps it is useful to explain the derivations above.
Parens is (close to) the present participle of parere, "to produce offspring". (There are many possible translations, but let me stick to this one here.) The regular participle would be pariens, but losing the i is hardly a far fetch, especially given the meanings of the words. The participle pariens is attested and seems to be used as a participle, but parens has taken up a life more independent of the verb.
Parturire is "to desire to produce offspring" or "to be about to produce offspring". The meaning "to be pregnant" is hardly a surprise. Derivations like this in -urire are formed from the perfect participle, which for parere is parit- or part-.
Parturitio is a noun for the action of parturire and has also the meaning "to give birth".
Partire means "to share or divide", from pars.
Partitio comes from partire just like parturitio from parturire.
Etymologies outside Latin
I am by no means an expert on this, so I will simply report what is on Wiktionary. Others better versed in etymology can probably give details. (Feel free to edit details in if you find that useful.)
- Wiktionary derives pater from PIE *ph₂tḗr.
- Wiktionary derives parere from PIE *perh₃-.
- Wiktionary derives pars from PIE *perH- or *per-.
The conclusion is that these three appear to be unrelated.