Consider two parallel reactions:
$$\ce{A -> B }\tag1$$ and $$\ce{A -> C}\tag2$$
What is the meaning of partial half life of the equation $(1)$ and that of equation $(2)$ with respect to above reactions?
It seems to me that it means the half life when one of these reactions are taking place individually, one-at-a-time, and not simultaneously. For example, partial half life of the equation $(1)$ is when half of $\ce{A}$ converts to only $\ce{B}$ and not to $\ce{C}$.
The partial reaction half-life relates to the reaction speed constant by the same way as for the "normal" reaction half-life.
If there are 2 parallel reactions of the first order, $\ce{A -> B}$ and $\ce{A -> C}$, and if there is the reaction rate for the former:
$$\frac{\mathrm{d}[\ce{B}]}{\mathrm{d}t}=k_{\ce{B}} \cdot [\ce{A}]$$
then the partial half-life w.r.t. $\ce{A -> B}$ is:
$$t_{1/2,\mathrm{B}}=\frac{\ln{2}}{k_{\ce{B}}}$$
Analogically the similar for the other reaction:
$$\frac{\mathrm{d}[\ce{C}]}{\mathrm{d}t}=k_{\ce{C}} \cdot [\ce{A}]$$
$$t_{1/2,\ce{C}}=\frac{\ln{2}}{k_{\ce{C}}}$$
The overall half-life w.r.t. $\ce{A -> X}$ is then
$$t_{1/2}=\frac{\ln{2}}{k_{\ce{B}}+k_{\ce{C}}}=\frac{1}{\frac 1{t_{1/2,\ce{B}}}+\frac 1{ t_{1/2,\ce{C}}}}$$
---- Responses to comments
The partial half-life is the extrapolated time after which all $\ce{A}$ would have decayed, if it had been decaying by the current and constant rate of given reaction and only by this reaction. But the main meaning is as a kind of reciprocal value of the reaction constant.
If we draw the chart of the partial reaction rate, then it's tangenta at $t=0$ will cross $x$-axis at the reaction partial half time. It is the same as if it was the only reaction. It does not say half of $\ce{A}$ decays in this partial half time.
If $\Delta t \ll \min{(t_{1/2})}$, then it does not matter it is just partial halftime, as we can neglect decay of the other parallel reactions.
