Isn't the parameter A denoting technological progress from Cobb-Douglas already included in alpha and beta?

The word technology might have slightly different meaning depending on a field. For example, Varian in his Microeconomic analysis (3ed) calls the whole production function technology. He literally writes on a first page of his book:

The simplest and most way to describe technology of a firm is the production function

However, in macroeconomics it is often equated to the $A$ parameter (see Romer Advanced Macroeconomics chapter 1 where technology growth and growth of $A$ for Cobb-Douglass function is used interchangeably throughout the chapter and also elsewhere in the book).

In the broader (Varian Meaning) you can consider alpha and beta parameters to be part of technology or determined by available technology (since whole production function is by definition in the Varian meaning determined by technology) but they are not technologies themselves per se (just their component). Rather they are output elasticities and in addition their sum tells you if there are increasing, decreasing or constant returns to scale, but technological progress does not necessarily mean that firms production will have different returns to scale or that output elasticity increases or changes in any way. Also, note these are virtually always considered to be constants.

In the macroeconomic sense they are not technology and the technology would be solely captured by $A$. In macro sense the $L^{alpha}K^{beta}$ would be some sort of ‘vanilla’ production which is then augmented by technology parameter $A$ (which could by the way also enter in a way that would augment only labor ($(AL)^{alpha}K^{beta}$) or only capital ($L^{alpha}(AK)^{beta}$) or also both ($AL^{alpha}K^{beta}$)).

So depending on which subfield you look at the terminology will differ, but I can’t recall ever seeing in the literature someone claiming that alpha and beta are the technology, at best you can argue they are determined by technology. Also depending on which field you specialize in you will want to follow the preferred terminology in the literature.