In germ cells telomerase enzyme is active and elongates telomeres, so youngsters are not born older than their parents due to telomeres.
Still, they do receive mutations, accumulated by their parents' germ-line cells, i.e. eggs and sperm. Most of the mutations occur upon division of the cell. Egg cells don't undergo many divisions - they are created in the course of embryonic development of a woman and then are spent one-by-one. At the same time, precursors of sperm cells divide constantly, thus accumulating mutations at a constant rate.
Got no time to look for a hyperlink, but I read that no matter what is the age of the mother, she gives 15 new single-nucleotide mutations on average to her offspring. Father gives about 10 mutations for each year of his life after reaching puberty, historically 70 on average. Out of those mutations most are neutral or silent (due to genetic code degeneracy), but on average ~1/70 leads to a change in aminoacid sequence of some protein, usually harmful.
So, although babies don't get older, but alas, they get uglier as generations pass :( Natural selection is what saves us from deterioration, so that the weakest are sentenced to childlessness.
Added a couple of hyperlinks as requested: decay of cognitive indicators of children with parent's age:
Another link: drosophila simulation of middle-class neighborhood (MCN) population, where each family has exactly 2 children, son and daughter, and no natural selection pressure is applied: such population rapidly deteriorates with fitness in wilderness decreasing by 2% per generation: