Let me preface by noting that I have a thorough understanding of DNA and it's use in genetic studies. That said, the above statement is pure nonsense.
Promiscuity verses monogamy has exactly zero to do with the passing down of mitochondrial and/or yDNA. The mother could be a complete and total slut who had a child with every man in the villages, or she could be totally monogamous, in either case however her mtDNA is still going to be "pure" as passed on to any and all of her children. Likewise the father could be a sexual ally-cat and have kids with dozens of wives or just one. His paternal or yDNA will be handed down to any and all of his sons, intact and traceable.
The rates of mutation in the two may vary, with the mtDNA doing so at a slower rate, but that's not what was articulated in the gibberish quoted above.
As evidenced by my DNA. My mitochondrial (maternal) line goes back 40-60 millenia, my yDNA only 24 when the protoCelts came out of the Iberian refugium as the glaciers melted and within a few generations populated Britain.