There have been some good answers here, but I think some information could be added.
Willi Hennig introduced modern phylogenetic systematics, which is sometimes conflicting with traditional taxonomy.
Hennig's important insight was that one should only use synapomorphies (shared derived traits) as the evidence for identifying relative recency of common ancestry within and among groups.
However, many groups, reptiles, fish, invertebrates, etc., are based on symplesiomorphies (shared ancestral traits).
The original question posted above can be addressed with the stem-crown group concept.
Considering the relationship of fossil taxa with living ones, we can ask the following question: do all fossil taxa fit into living clades?
Crocodiles and birds are an excellent example to address this question:
From a cladistic perspective, members of each of the two clades have three types of traits: symplesiomorphies (shared ancestral), synapomorphies (shared derived), and autapomorphies (unique traits).
Where would these traits appear on a phylogenetic tree?
Let us have a look at a bird with a very distinctive bill, the Shoebill (Wikipedia):
This can be considered a unique trait, thus an autapomorphy.
The feathers of the shoebill is a trait it shares with other birds, thus a synapomorphy.
While the shoebill's legs is a trait shared with crocodiles (the sister clade of birds) and many other vertebrates and therefore is a symplesiomorphy.
Thus, autapomorphies appear within the clade, synapomorphies are traits primitively shared by all the members of the clade of interest, but not by members of its sister clade. They are shared but relatively new traits: these traits must have appeared further down the clade, but not below the split between the two sister clades.
Symplesiomorphies are traits shared by the sister clade, or even more distant clades. These traits must have appeared deeper down the tree.
Another question: what sort of organism was the last common ancestor of birds and crocodiles? Was it a bird, a crocodile, or neither? What traits would it have?
By definition of our question, these two groups (crocodiles and birds) are both monophyletic, therefore, one did not give rise to the other. It did not have any of the synapomorphies or autapomorphies in either clade, only their shared symplesiomorphies, and therefore did not lie in either modern clade.
By definition, the organisms that make up the lineage of birds, appearing after the split between the clades of crocodiles and birds, but apart from living birds, must be extinct. We call this the stem group of birds.
The last common ancestor of all living birds, plus all its descendants make up the crown-group of birds. The total group of birds is the crown group and stem group together.
Stem and crown groups are hierarchical, and some crown group taxa have gone extinct, for example, the dodo. The dodo, therefore, belongs in the stem group of pigeons and doves.
By definition, all stem groups are paraphyletic, as they give rise to their crown group. All fossils fall into one or another stem group.
The original question can thus be answered appropriately: dinosaurs are stem-birds and belong in the total group of birds.