Was there a D to TH sound change in English?

Solution 1:

Grimm's law is at work here.

Grimm's law consists of three parts which form consecutive phases in the sense of a chain shift. The phases are usually constructed as follows:

Proto-Indo-European voiceless stops change into voiceless fricatives. Proto-Indo-European voiced stops become voiceless stops. Proto-Indo-European voiced aspirated stops become voiced stops or fricatives (as allophones). This chain shift (in the order 3,2,1) can be abstractly represented as:

bʰ → b → p → f

dʰ → d → t → θ

gʰ → g → k → x

gʷʰ → gʷ → kʷ → xʷ

Here each sound moves one position to the right to take on its new sound value.

(Source: Grimm's Law at Wikipedia)

Consider the chain shift in bold. This phenomenon is at work in the Latin dental to English tooth and in the examples cited by the OP, from Latin pater to OE fæder to English father.