Regardless of which is more commonly used, I need to point out a potential issue with parallelism, which you've alluded to in your question. (I will bypass the issue of which pronoun to use, with or to, and continue to use the same one as in your example sentences.)
Agree is in the present tense, while agreed is in the past tense. Because of the use of have, read is in the past tense.
Normally when we form a sentence with two verbs, and elide the same auxiliary verb from the second verb, we make sure that both are in the same tense. This isn't always essential, but it's a common consideration.
I would say that any potential issue with your sentence would be one of style more than strict grammar.
I have read [past tense] and agree with [present tense] the terms and conditions.
The normal way of reading such a sentence is to apply the assumed missing information from the first part to the second part:
✘ I have read and [I have] agree with the terms and conditions.
That, however, is wrong. As you've said, it can be explicitly fixed by repeating the use of the subject (without the auxiliary verb), in order to make the conjugation clear:
I have read and I agree with the terms and conditions.
However, even though that is now grammatical, it's still slightly odd to be mixing verb tenses in this way.
Typically, they would both be in the past tense or the present tense:
✔ I read and agree with the terms and conditions.
✔ I have read and agreed with the terms and conditions.
In this case, even though only slightly strange, the mixing of verb tenses isn't really a problem.
But it would be a clear problem if the sentence used different words.
✔ I have understood and signed the document.
✔ I understand and sign the document.
✘ I understood and sign the document.
✘ I understood and I sign the document.
And to compare it to your sentence:
❔ I have read and agree with the terms and conditions.
❔ I have read and I agree with the terms and conditions.
What makes this interesting is that while we reject understood and sign because of the mix of past and present tenses, your exactly constructed sentence (but with read and agree) may arguably be idiomatic.
(Why your version doesn't sound immediately wrong may have something to do with the fact that the past tense and present tense of read are spelled identically, even though they are pronounced differently. Or it may have something to do with agree being a condition that continues from the past into the present, rather than a one-time event like sign.)
Still, if I force myself to view your sentence in the same way as the other sentence, it seems less acceptable to me. So, from this strict perspective, if you want to be consistent about this particular sentence construction, it would be best to avoid mixing the tenses.