Can I losslessly decompose this table?
Normalization is based on functional dependencies. Functional dependencies have to do with semantics; they have to do with what the data means. When you simplify a real-world problem to the level of "parent_id, child_id, date", and you don't include any sample data, you really limit how much help a conscientious database designer can give you.

The fact that you have a key {child_id, parent_id, date} in one table, and that you have (it seems) a unique pair {child_id, parent_id} in the child table doesn't necessarily mean that part of the combination is redundant. It might mean that in the table that has {child_id, parent_id, date} as the primary key, the pair of attributes {child_id, parent_id} ought to reference the child table in the first place. 

If that's the case, you might use FOREIGN KEY (child_id, parent_id) REFERENCES child (child_id, parent_id). To do that, you need a UNIQUE constraint on the pair of columns (child_id, parent_id) in the table "child", which shouldn't be a problem if child_id is its primary key.

But there's no way to tell without knowing what the data means, and you're the only one in this thread who knows that. (But we'd be happy to let you explain it to us.)

As far as the original table is concerned, you seem to be saying that child_id -> parent_id. If that's the case, why is parent_id in the original table in the first place? Why isn't the key just (child_id, date), with a foreign key reference to the "child" table? It looks to me like the kind of redundancy you're talking about might be resolved by dropping the column "parent_id".

SQL DDL and sample data in the form of INSERT statements helps us help you. DDL and INSERT statements are more precise than descriptions.