No, that sounds strange, for a few reasons - one of which you've mentioned.
"Today Peter bought a bicycle, which he rode to the pool."
"Today Peter bought a bicycle, which he has ridden to the pool."
The use of 'today' actually counts as a specific time in the past in this context, because, as you say, it occurred before something else you are relating in the past tense. It means "earlier today" - past.
"Peter has a bought a bicycle, which he has ridden to the pool." OK, since they are both in present perfect. One imagines a person at the pool saying this, and there are conceptualised as one event for the purpose of tenses.
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